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BOOK
:matics
: TWO
clo
MODERN JUNIOR MATHEMATICS
Book Two
Suggestions
The prefaoe will make olear Miss Sugla^s
basic Ideas on doing and thinking in terms
of mathematical relations •
Think of all the directions as written for
the pupil to read and follow^
Consider the liireness of a class in mathe-
matics using the equipment listed on page
xiii^
Begin with Pa^e 1. Follow the thought »
step hf step, for a chapter or two» See
how easily the subject-matter is dereloped
and logical thinking processes stimulated «
Interesting features are found on -the fol-
lowing pages:
pp lOD— 30^ Section D
101, Section B
196 » 225
r
Illll HilEIIBJII Ulll
SCHOOL OF EDUCATION
LIBRARY
TEXTBOOK COLLECTION
GIFT OF
THE PUBLISHERS
STANFORD V^p/ UNIVERSITY
LIBRARIES
Tan ssiBinji! fjuil
SCHOOL OF EDUCATION
LIBRARY
TEXTBOOK COLLECTION
GIFT OF
THE PUBLISHERS
STANFORD ^^g^ UNIVERSITY
LIBRARIES
MODERN JUNIOR MATHEMATICS
BOOK TWO
MODERN
JUNIOR MATHEMATICS
BOOK TWO
BY
MARIE CTJGLE
ASSISTANT SUPEBINTEnBeNT OF SCHOOLS
COLUMBUS, OHIO
THE GREGG PUBLISHING COMPANY
NEW YORK CHICAGO BOSTON SAN FRANCISCO
Liverpool
'^ -•
615106
C
COPYRIGHT, 1920, BY THE
GREGG PUBLISHING COMPANY
A 5 a
PREFACE
Until recently upper elementary and high school
work in mathematics was planned for the pupil who
was expected to continue it in the university. Al-
though logical, its arrangement was neither psycho-
logical nor pedagogical, but some progress has been
made recently in adapting the study to the needs and
abiUties of pupils.
In the junior high or intermediate school, work in
mathematics in the seventh, eighth, and ninth grades
should be complete in itself and at the same time pre-
paratory to senior high school work. No effort should
be made to '' finish'' arithmetic in the eighth grade
and algebra in the ninth, while denying the child the
interest and beauty that lie in geometry and trigo-
nometry until his taste for mathematics has been de-
stroyed. Nor will alternate bits of formal algebra, I
geometry, and trigonometry solve the problem. The'
result is a mastery of none and a confusion of all.
Experience has proved that the necessary elements
of arithmetic can be taught and certain definite skill
developed in the first six grades. In the seventh grade
business applications of arithmetic with the simplest
elements of bookkeeping should be given. In the
eighth grade, mensuration should be taught experi-
mentally or through observational geometry, and '
through that, in a natural and meaivm^Wi ^^ ^ '^^^
• • •
m
iv PREFACE
very beginnings of algebra. Optional courses should
be offered in the ninth grade.
Experience in junior high schools has shown that
much of the content and the whole of the organization
of subject matter must be changed to make the course
of study fit the needs of the pupils.
The definite aims of this study are :
^1. To extend the pupil's knowledge of arithmetic
through its practical applications in mensuration.
>-- 2. To train the hand to use the simple drawing in-
stnmients.
^ 3. To familiarize the pupil with common geometric
forms.
" 4. To train him to see geometric forms in nature
and in the various buildings and other structures in
his surroundmgs, and to appreciate their use in de-
sign.
5. Through experiment and observation to develop
the formulas of mensuration.
6. Through a continued study of formulas to in-
troduce general niunber in a natural way that will
give algebraic expressions such a meaning to the pupil
that he will use them as convenient and practical
tools.
7. To permit a pupil to live so continuously in the
atmosphere of geometry that he may be enabled to
think naturally and without confusion, in its terms,
about its relations.
This book is planned for a year's work in the eighth
grade, with the idea that the pupil should advance
slowly by doing and thinking for himself. If neces-
sary it can be condensed into one semester's work. It
PREFACE V
should always precede and form an introduction to the
more fonnal algebra and geometry.
The nucleus of this book grew out of the author^s
experience as an instructor in mathematics in a large
city high school, in teaching the first book of plane X
geometry without a text. Four years ago an outline
of it was given by the author to the teachers imder
her supervision because no suitable text in experi-
mental geometry could be found. By way of further
suggestion, one or two topics were expanded in more
detail and discussed with the teachers. This book is
the result of their urgent request for more. Since
then, over a score of teachers have used the outline,
and their imanimous opinion is that the pupils take
an increased deUght in mathematics of this kind. The
author was in a position to observe the effects of teach-
ing it in various types of schools and found it most
gratifying.
The author is very much indebted to Dr. John H.
Francis, Superintendent of Schools, Colimibus, Ohio,
for reading the manuscript and for giving helpful sug-
gestions. She hereby acknowledges her indebtedness
also to Miss Meta Philbrick of Mt. Vernon Inter-
mediate School and to Miss Amy Preston of Roosevelt
Intermediate School, Colimibus, for their cooperation
and assistance in gathering problems and program
material.
Makie Gugle
CONTENTS
BOOK TWO
PAGE
Preface iii
Equipment for Mathematics Departments xiii
CHAPTER ONE
FORM AND MEASUREMENT
A. Form 1
I. Blocks 2
B. Measurement 3
I. Perimeter 3
•
CHAPTER TWO
SURFACES OF COMMON SOLIDS
A. Study of Cubes and Oblongs 7
I. Squares and Rectangles 7
II. Angles 7
B. Other Quadrilaterals 11
I. Parallelograms 11
C. Measurement of Area 13
I. Area of Squares and Rectangles 13
II. Surface of Cube and Oblong 15
D. Square Root 19
I. Study of Numbers and Factors 19
II. Finding Square Root by Prime Factor Method .... 21
III. Finding Square Root by Mechanical Method. ..... 22
IV. Finding Square Root by Inspection 29
V. Review Problems 30
vii
viii CONTENTS
PAQB
CHAPTER THREE
A STUDY OF TRIANGLES
A. Triangles from Squares 31
I. Reading of Angles 31
II. The Sum of the Angles of a Triangle 32
III. Complementary Angles 34
IV. Intenor and Exterior Angles of Polygons 35
B. Triangles from Rectangles 36
C. Classification of Triangles 37
I. According to Size of Angles 37
II. According to Equality of Sides 37
D. Area of Triangles 38
I. Area of Right Triangle* .... * 38
II. Area of Scalene Triangle 39
CHAPTER FOUR
CONSTRUCTIONS
A. Constructions of Triangles 43
B. Construction of Lines 47
I. To draw Perpendicular Lines 47
II. To Draw Lines of Bisection 51
CHAPTER FIVE
FURTHER STUDY OF TRIANGLES
A. Important Lines in Tioangles 53
I. In Isosceles Triangles 53
II. In Scalene Triangles 54
III. Designs 56
B. The Right Triangle 60
I. Its Properties and Practical Uses 60
CHAPTER SIX
PARALLEL LINES
A. Meaning of Parallel Lines 69
B. Construction of Parallel Lines 69
CONTENTS ix
PAGE
I. With Protractor 69
II. With T Square 70
III. With Drawing Triangle 70
IV. With Ruler and Compass 71
C. Parallels Cut by a Transversal 72
I. Angles Made by a Transversal 73
II, Vertical Angles 73
III. Supplementary Angles 73
D. Practical Use op Parallels 75
CHAPTER SEVEN
QUADRILATERALS
A. Construction of Quadrilaterals 79
B. The Square 80
I. Relation of Lines and Angles 80
C. The Rectangle 82
I. Relation of Lines and Angles 82
D. The Parallelogram 83
I. Area. 83
II. Relation of Lines and Angles 85
III. The Rhombus 86
E. Summary 86
CHAPTER EIGHT
MEASURED AND UNMEASURED LINES
A. Problems from Rectangles 87
I. Parentheses 87
II. Ratio 90
III. Graphs of Ratios 96
B. Problems with Unmeasured Lines 101
I. Perimeters and Areas 101
II. Finding One Dimension 119
III. Problems of Finding Dimensions from Areas 129
CHAPTER NINE
REVIEW PROBLEMS
A. Problems on Parallelograms 134
B. Problems on Triangles 136
X CONTENTS
PAGE
CHAPTER TEN
SIMILAR FIGURES
A. SiMiiAR Rectangles 142
B. Similar Parallelograms 143
C. Some Practical Uses of Similar Figures 145
D. Similar Tiuangles 145
E. The Use of the Quadrant and Sextant : 150
I. Drawings from Bettinus 150
CHAPTER ELEVEN
POLYGONS
A. Trapezoids 161
B. Other Polygons 165
C. Construction of Regular Polygons 167
I. The Hexagon 167
II. The Octagon 171
III. The Pentagon 174
CHAPTER TWELVE
CYLINDERS AND CIRCLES
A. Cylinders 179
B. Circles 181
I. Circumference of a Circle 181
' II. Area of a Circle 185
C. Surface of a Cylinder 187
D. History of Pi 188
E. Problems — Cylinders and Circles 190
CHAPTER THIRTEEN
VOLUME
A. Volume of a Prism 194
B. Volume of a Cylinder ^ 195
C. Problems 196
CHAPTER FOURTEEN
REVIEW OF FORMULAS
A. Translation of Formulas 198
B, Formulas Given in the Supplement 198
CONTENTS xi
PAGE
SUPPLEMENTARY TOPICS
A. Inspection Method op Finding Square Root 199
B. Hero 's Formula for Finding the Area of a Triangle . . 200
C. Less Common Solids 202
I. The Pyramid 202
II. The Cone 204
III. The Sphere 205
IV. Patterns for SoUds 206
MISCELLANEOUS PROBLEMS 209
REVIEW QUESTIONS BY CHAPTERS . . .221
SYMBOLS AND ABBREVIATIONS 225
APPENDIX
A. Mathematics Clubs 227
B. Tables 231
C. A Protractor and its Use 235
INDEX 237
EQUIPMENT FOR MATHEMATICS
DEPARTMENTS
IN INTERMEDIATE AND HIGH SCHOOLS
1. Graph chart 1 per room
2. Yard or meter sticks 2 dozen per room
3. Balls of cord 2 balls per room
4. Parallel rulers 1 per room
5. Protractor for blackboard .... 1 per room
6. 100-ft. tape line 1 per room
7. Set of mensuration blocks .... 1 per room
8. Scissors 2 dozen per room
9. Equation balance
10. 2 large wooden triangles .... 1 set per room
(90-60-30 and 9045-45 degrees)
11. Slated globe (8'' or 12" mounted
and detachable) 1 per room
12. Adjustable geometric models 1 set per teacher of
solid geometry
13. Inexpensive transit 1 per building
14. Supply of cardboard, paste, and heavy paper
15. Books 1 copy per building
(a) History of Mathemathics — Ball
(b) Scrapbook of Mathematics — White
(c) Mathematical Wrinkles — ^Jones
(d) Mathematical Recreations — Ball
(e) Flatland — Abbott
(f) Magic Squares and Cubes — Andrews
(g) The Hindu-Arabic Numerals — Smith & Kar-
pinski
(h) Number Stories of Long Ago — Smith
... •
xm
xiv EQUIPMENT
EQUIPMENT FOR EACH PUPIL
1. Combination ruler and protractor
(of transparent celluloid marked in inches and centi-
meters; some inches divided into sixteenths and some
into tenths)
2. Compass that uses a pencil
3. Graph paper
4. Two right triangles — (90-60-30 and 90-45-45 degrees)
MODEKN
JUNIOE MATHEMATICS
BOOK TWO
CHAPTER ONE
FORM AND MEASUREMENT
A. FORM
Heretofore your interest in mathematics bas
centered around computation, especially in making
with accuracy and" speed the calculations used in
everyday busmess." In order to retain the skill you
have developed, it will be necessary for you to practice.
But there is a great deal more in mathematics than
mere calculation with figures. This is a very impor-
tant and necessary part of all mathematics, although
it is the newest part, for the figures which make our
nmnbers were not known in Europe until the thir-
teenth century. Nevertheless fifteen himdred and
more years before, there hved some of the most famous
mathematicians of all times. A knowledge of the
mathematics developed by these men enables us to
survey our land, to buUd our houses, buildings, bridges,
and ships, to design our furniture and our art, and to
make our patterns for various kinds of manufactures.
We live in a world of people no two of whom are
exactly alike, although all faces have the same featwxo.^
1
2 MODERN JUNIOR MATHEMATICS
of forehead, nose, eyes, ears, mouth, and chin. We
also Uve in a world of things which, though they are
many and varied, have a few common features or forms.
I. Blocks
The cube and oblong blocks are two of these forms
which every one knows. But, although they are so
Cube Oblong Blocks
familiar, most people cannot answer offhand a few
simple questions about them.
Without examining these blocks further, try to
answer the following questions without hesitation:
1. Name different objects around you that have these
shapes.
2. How many dimensions has each block?
3. What are these dimensions?
4. How many faces has each block?
5. How many edges has each?
6. How many comers has each?
7. (a) How does the position of any one face compare
with the one opposite it?
(b) Two such opposite faces or surfaces that are
everywhere the same distance apart are parallel.
(c) The word parallel is derived from two Greek words
which mean beside one another, A mathema-
FORM AND MEASUREMENT 3
tician writes parallel thus, ||. This sign, ||,
may mean parallel or is parallel to. A small s
beside it makes it plural. ||s means parallels.
8. What is the shape of the faces of a cube? of an oblong?
9. How many edges of a cube are equal?
10. How many edges of an oblong are equal?
11. Compare the lengths of the opposite edges of any
one face; the position.
B. MEASUREMENT
I. Perimeter
1. (a) The word perimeter comes from two Greek words
which mean the measure around. The perim-
eter of any object is the measure aroimd it.
(b) You are familiar with the word meter in the terms
. gas meter, speedometer (speed measure), and
thermometer (heat measure). ^^
2. (a) Find the perimeter of a face of the cube.
(b) How many edges must you measure to find the
perimeter of one face?
(c) Is the following statement true? If so, why?
The perimeter of a face of a cube or of -a square is
equ^l to four times the length of one edge.
3. (a) Is this true for any face of the oblong? Why?
(b) How many edges of the oblong must you meas-
ure to know the perimeter of one of its faces?
4. (a) To distinguish certain faces of a cube or of an
, oblong from others, definite names are given.
j (b) The face on which the object stands is called the
1 base or the lower base. The opposite one, or
J the top, is the upper base.
(c) The faces around the sides are called lateral
faces. Lateral comes from a Latin word whick
means side.
4 MODERN JUNIOR MATHEMATICS
6. (a) Can you find the perimeter of the base of an ob-
long from the length and width only? How?
(b) Show that the perimeter of the base of an oblong
or of a rectangle is equal to two times the sum
of its length and mdth.
6. (a) Find the perimeter of a cube whose edge is 3
inches; 7 inches; SJ inches,
(b) Find the perimeter of the base of an oblong that
is 6 inches long and 4 inches wide; of one that
is 8 feet long and 3^ feet wide; of one that is
5^ inches by 2J inches.
7. In the statements, sections 13 (c) and 16 (b), let the
first letters of the main words be used in place of the words,
thus:
(a) For square, P = 4 x e
(b) For rectangle, P =2 x (I +w)
(c) By putting parentheses ( ) around Z + ty, we
show that we must find the sum before multi-
plying by 2.
8. (a) Such a short way of writing a statement is a
formula.
(b) Carpenters, surveyors, engineers, and many other
craftsmen find it very convenient to use formu-
las for several reasons:
1. Because formulas take much less writing and arc
more quickly read than the English statement in full.
2. Because such formulas are true for all rectangles
and squares.
9. (a) If the edge of a cube or e = 3 in., then by put-
ting 3 in the place of e in the formula, we have
for the cube or square, P = 4 x «
= 4 x3
= 12 inches
FORM AND MEASUREMENT 5
(b) If a rectangle is four inches long and three inches
wide, then we have
for the rectangle, P = 2 x (I -{-w)
= 2 X (4 + 3)
= 2 x7
= 14 inches
(c) This process is called substituting numerical values
for the letters in a formula.
(d) Instead of writing out the words "for a square"
or "for a rectangle" to show which particular
perimeter we mean, we may draw a little square
or a httle rectangle directly after the letter P.
Pq means perimeter of a square.
Pa means perimeter of a rectangle.
10. (a) We have seen that
Pn = e +e +e +e
= 4 X e
Therefore, 4 x e means that e has been added to
itself four times, or e has been multiphed by 4.
It is shown in
12 =3+3+3+3
= 4 x3
a
(b) A number used (as the 4) to show how many
times another number (as the e or 3) has been
added to itself is a coefficient.
Since multiplication is only shortened addition,
the coefficient shows that the other part of the
expression is to be multiplied by the coefficient.
(c) When we multiply two arithmetical nmnbers to-
gether we have to use the times sign (x), as
4x3; but when one number is a letter, as
4 X e, we may omit the time!& ^y*^ ^tA ^^r62^^
it 4 e (read "four e"V
6 MODERN JUNIOR MATHEMATICS
(d) If the sign were omitted between the 4 and 3,
what would the number mean?
(e) Instead of using the cross (x) to indicate multi-
phcation, we may use a dot placed between the
numbers and above the hne of writing. If it
were placed on the line of writing, with what
might it be confused?
3-7 means 3 times 7, just as 3 x 7.
The use of the dot avoids the confusion of the
cross with the letter x.
What is the value of 2.2.3-7? of 3.4-7.2.5?
11. What are the coefficients of the following?
5 e, 6 cwt., 10 T., 2§ doz., 3 lb., 2 ft., 25^.
12. (a) What do C and M stand for in Roman numerals?
(b) How many sheets of letterheads are there in 2 M?
In4C?
(c) How many are there in 2 M + 4 C?
13. How many pounds of hay are in 7| T.?
14. In two mows there are 2\ T. and 3f T. of hay, re-
spectively. How many T. are in both? how many pounds?
16. By using the formula for a square, find the perimeter
of a cube whose edge is (a) 4 in.; (b) 1\ in.; (c) f ft.
16. By using the formula for a rectangle, find the perim-
eter of the base of an oblong whose length and width,
respectively, are (a) 7 in. and 5 in. ; (b) 6^ in. and 4^ in.
17. (a) Estimate the length and width of your desk and
find the perimeter,
(b) Measure with your ruler and find the perimeter
more exactly.
18. Estimate and then measure the perimeter of (a) your
book; (b) your school room; (c) your teacher's desk;
(d) a picture on the wall; (e) a door; (f) a window;
(g) any other square or rectangular objects that may be
measured convemently.
CHAPTER TWO
SURFACES OF COMMON SOLIDS
A. STUDY OF CUBES AND OBLONGS
I. Squares and Rectangles
1. Paper ruled into small squares is called squared
paper. Each pupil should have a supply of it.
2. On squared paper draw a picture of one face of a
cube; of an oblong.
3. We have observed that the opposite edges are par-
allel. For convenience, let us name the four comers with
A BE F
DON Q
different letters, so that we may tell which comer we are
talking about.
4. (a) Let us call the side between the comers at A
and Bj the line AB.
(b) Read the names of the other sides.
6. (a) Two sides that meet at the same comer or point
are called adjacent sides; as, AB and AD,
(b) The word adjacent means lying next to,
(c) Name the pairs of adjacent sides in each figure.
(d) Name the pairs of opposite sides in each.
n. Angles
1. (a) Whenever two lines are drawn from the same point
an opening is formed called an angle.
7
8
MODERN JUNIOR MATHEMATICS
(b) Angle means comer.
(c) An angle is the amount of opening between two
straight Unes that meet.
(d) The size of the angle increases as the lines separate
from each other.
The hands of a clock or watch always form an angle
which is constantly changing in size.
:i
The hands on the face of a large clock make the same
angle at 2 o'clock as the hands of a small watch. The
length of the hands has nothing to do with the size of
the angle. The size depends solely on how far apart the
hands are.
2. (a) At 3 o'clock and 9 o'clock the angles which the
hands form are right angles.
(b) At 6 o'clock the hands form a straight hne and
the angle becomes a straight angle.
3. (a) Fold a sheet of paper smoothly.
(b) Fold it again, putting the two edges together
carefully.
STUDY OF CUBES AND OBLONGS 9
(c) The four angles or quarters are now equal, for
they exactly fit each other.
(d) Unfold the paper. The two creases or Unes cross
each other, so that the four angles are equal
and are right angles.
(e) If one line meets another line so that the two
angles are equal, the angles are called right
angles,
(f) Two lines that meet so as to make equal angles
or right angles are perpendicular to each other.
4. (a) Use this folded paper to measure the angles of
the square and rectangle.
(b) What kind of angles are they?
(c) What is the smu of all the angles in each figure?
(d) The word rectangle means right angle. Therefore,
if a four-sided figure has all its angles right
angles, it is a rectangle.
(e) The sign for right angle is rt. Z ; in the plural
it is rt. A.
(f) The sign for perpendicular is ±, or -b for the
plural. Sometimes J. means is perpendicular
tOf as AB± CD means that the line AB is per-
pendicular to the line CD,
6. (a) Just as a foot ruler is divided into twelve smaller
parts caUed inches, so a right angle is divided
into ninety smaller angles called degrees,
(b) A right angle is said to contain ninety degrees
(90°).
6. (a) Unfold your angle paper once.
(b) How does this angle compare with that made by
the hands of a clock at 6 o^clock?
(c) What is the name of this angle?
(d) How many right angles are in it?
(e) How many degrees are in a attav^X, ^\v^<^,
7. To draw right angles and peipendicviJL^x \fflL^'a»> ^^st-
10
MODERN JUNIOR MATHEMATICS
penters use a steel instrument called a square or T square,
depending on whether its shape is that of the letter L or T.
1
J
Carpenter's Square
T Square
8. (a) At a quarter of nine o'clock the hands are aUnost
together. At 9 o'clock they form a right angle.
In that fifteen minutes many different sized
angles were formed, but all were less than b.
right angle.
(b) Any angle whose size is less than a right angle is
an acute angle.
(c) Acute means sharp,
(d) Draw an acute angle and see why it is properly
named a "sharp" angle.
Right Angles
Acute Angles
Obtuse Angles
9. (a) What is the size of the angle made by the hands
of a clock at 9 o'clock?
(b) At 9.15 it is almost but not quite a straight angle.
(c) In that fifteen minutes many different sized angles
were formed, but all were greater than a right
angle and less than a straigjat aii^^.
OTHER QUADRILATERALS
11
10.
(d) Any angle that is greater than a right angle and
less than a straight angle is an ohtuse angle,
(e) Ohtuse means hlunt,
(f) Draw an obtuse angle and see why it is properly
named a " blunt ^' angle.
(a) A carpenter tests his squares, that is, he sees
whether they are true right angles or not, by
placing two of them together on a straight
edge, as shown in the figure.
(b) If the two edges exactly fit when thus placed, the
outside angles are true right angles. Why?
(c) How may the inside angle of a square be tested?
I
1
L
]
11. What kind of angle do the hands of a clock make at
3 o'clock?
12. Do they make the same kind at a quarter past 12?
13. At what times do they make right angles?
14. Do they make a straight angle at 12.30?
16. What kind of angle is made by the hands of a clock
at 3.30, 3.35, 1.00, 1.30, 4.30, 8.00, 8.55, 8.40 o^clock?
B. OTHER QUADRILATERALS
I. Parallelograms
1. (a) How many sides has a square?
(b) Are all of the sides of a squats ^o^^Vl
(c) Are the opposite sides paiaWeYl
12 MODERN JUNIOR MATHEMATICS
(d) Are all of the angles right angles?
(e) Are aU of these features necessary for a square?
(f) Name all the necessary features of a square.
2. (a) Try makmg four-sided figures with only two of the
other features, as given in (6), (c), and (d).
(b) Make a four-sided figure which has only the
feature (c).
3. (a) Which features are found and which are lacking in
the following figures?
Rectangle Rhombus Rhomboid
(b) All of these figures are parallelograms because each
has two pairs of parallel sides.
(c) Parallelogram means parallel drawivvg.
(d) The sign for parallelogram is O. For the plural it
is UJ,
(e) Fill in these blanks to make a correct definition:
A parallelogram is a figure inclosed by — pairs of
— lines.
(f) Is a square a parallelogram?
(g) Is every square a rectangle?
(h) Is every rectangle a square?
(i) We always give the special name of square, rectangle,
or rhombus to each of the particular kinds of
parallelograms, but seldom use the name rhom-
hoid. When we speak of a parallelogram we
mean the general form or rhomboid.
4. (a) How many objects can you find having these
shapes?
MEASUREMENT OF AREA
13
(b) What is the shape of your schoolroom? of your
school yard? of the pages of your book? of the
door?
6. Below is a map of a section of a city. What are the
shapes of the various city blocks?
JV
"\r\\
\
r
C. MEASUREMENT OF AREA
I. Area of Squares and Rectangles
1. On squared paper draw a picture of one face of a
cube whose edge is one inch. This picture is called a square
inch and is used to measure surface or area.
(If metric units are desired, do the same with one
centimeter.)
2. Draw a square whose edge is 2 inches. Count the
niuuber of square inches in it. This number is called the
measure of the area or simply the area of the square.
3. Draw squares with different edges and count the
square units in area.
MODERN JUNIOR MATHEMATICS
B
A
Area of square A -2-2 or 2^ or 4 -1 sg. in. = 4 8q. in.
" " square B = 4-4 or 4' or 16-1 sq. in. = 16 sq. in.
" " square C = 5-5 or 5' or 25-1 sq. in. — 25 sq. in.
4. By measuring one edge of a square, how may you
find the number of square units in its area?
6. On squared paper, draw several rectangles and count
the square units in their areas.
Area of rectangle D = 4-3 or 12-1 sq. in. =12 sq. in.
" " " E = 5-3 or 15-1 sq. in. = 15 sq. in.
" " " F =7-5or35-l sq. in. = 35sq. in.
6, (a) Thus we find that the area or surface of a rect-
angle or of a square equals the length Umes the
vddth.
(b) In this statement let the first letters of the main
words be used in their places. Then we have
MEASUREMENT OF AREA
15
7. (a) What do we call such a statement?
(b) Does this statement hold true for all rectangles?
8. (a) Measure your desk, your book, and your school
room and find their areas,
(b) What units will you use for measuring each of
these?
9. (a) In what kind of rectangle is the length equal to
the width?
(b) If e stands for the edge of a cube, we may say,
area of one surface = edge x edge,
or
Sq = « • « = e^
10. If the edge of a cube is 3 inches, then
Sn = 3-3 = 32 = %sq. in.
11. Find the area of a face if the edge of cube is 6 in.;
4| in.
n. Surface of Cube and Oblong
1. Imagine the" surface of the cube as being a very
thin covering that may be peeled ofif as an onion skin. Run
I \
v
a knife down one edge and peel off that face without detach-
ing it. Do the same for the opposite face. Then peel off
the rest in one piece. Make a pattenv ol AiJt^ ^vs^^a^fc ^^
16
MODERN JUNIOR MATHEMATICS
stiff paper. Put on necessary flaps, cut out, paste, and put
tc^ether.
2. Do the eame for the oblong.
3. Another name for this oblong is parcUlelapiped. A
cube is also a parallelopiped, but all of its edges are equal.
4. Make two pattemsj one for an oblong with square
baees and one with rectangular bases.
I
n
c
/
h
d
b
6, (a) How many faces has a cube?
(b) How do the different faces compare in size and
(c) How would you find the total surface, that is,
the sum of the areas of all of the faces?
(d) The total surface of cube = 6 X area of one face,
Forcube^Tot. S=6e*
If the edge is 3 in., then
Tot. S - 6-3'
= 54 sq. in.
MEASUREMENT OF AREA 17
6. How many faces of a cube are lateral faces? Can you
see the reason for the following formula?
For cube, Lat. /S = 4 e^
7. (a) How many different dimensions has the first
oblong, A1 the second, Bf (b) Name them.
(c) What is the shape of the pattern of the lateral
surface?
(d) What is the total length of this rectangle?
(e) What dimension of the oblong is the width of this
rectangle?
(f) The formula for the lateral surface is
Lat. S =2-(Z +w?)./i
or, = 2/i(Z + w)
(g) Put this formula into a complete EngUsh sentence
and see if it is true for both figures,
(h) Can you think of a practical problem in which
you would want to find the lateral area only?
Would the painting of a fence be such a problem?
(i) How many bases has an oblong?
(j) How can you find the area of each?
(k) How can you find the total area of the parallelo-
piped?
(1) Lat. S=2.(Z + ti;).fe or Lat. S = 2 fe(Z + w?)
2 Bases, S = 2d'W
Bases, >S = 2 Zii;
Tot. & = sum of all parts
Tot. & = sum of all
parts
8. Find the total area of a chest if its length is 10 ft., its
width 4 ft., and its height 3 ft.
&olvii(m.
Lat. S =2 fe(Z + w?) = 2-3(10 + 4)
= 2. 3. 14 =84 sq.ft.
2 Bases, & ^2lw = 2»10'4 = 80 sq. ft.
Total /S = m"^^^.
18 MODERN JUNIOR MATHEMATICS
9. Another formula for total surface is,
Tot. S = 2(lw +lh + wh)
= 2(10-4 +10-3 +4-3)
= 2(40 +30 + 12)
= 2(82)
= 164 sq. ft.
Show why this formula is correct.
10. The second method is more convenient when only the
total area is desired. If the area of the lateral surface and
one base only Ls required, the first method is the better.
11. A flower box is 8 ft. long, 10 in. wide, and 8 in. high,
inside measurement. How many square feet of tin will be
required to line the box? No allowance is made for waste.
12. How many square yards of cement pavement are
needed for a walk 4^ ft. wide in front of a lot 50 ft. wide?
13. (a) An athletic field is inclosed by a board fence 10
ft. high. The field is 600 ft. X 400 ft. One
gallon of paint covers 250 sq. ft. with two coats.
How many gallons will be needed to paint the
fence?
(b) A painter calls 100 square feet a square. If he
can paint a square of fence in one hour, what will
be the cost of labor in painting the fence? In-
quire of a painter the scale of wages.
(c) What will be the total cost?
14. (a) How many square feet are in the side walls of
your school room?
(b) Find their area with the doors and windows taken
out.
15. (a) A house is 40 ft. X 32 ft. X 18 ft. How many
squares in its surface? (Use the nearest
integer.)
(b) Allow 14 squares for the dormer windows, cornice,
and porch. How many gallons of paint are
SQUARE ROOT
19
needed? (No aUowance is made for doors aad
windows because they take more paint than
flat surfaces.)
(c) Find the cost of the paint at $4.25 per gal.
(d) It takes a painter two hours to paint a square of
a house. Find the cost of labor at $.70 per
hour. Find the total cost.
(e) How many days are needed for the work?
(f) What does the painter earn in an 8-hour day?
in a week?
D. SQUARE ROOT
L Studj of ITiimberG and Factors
1. (a) On squared paper draw a square iuclosii^ 9 sq.
in.
(b) Draw one inclosing 25 sq.
in.
(c) How loi^ is the edge of
each?
2. Since the area of a square is
found by multiplying two equal
numbers tc^ther, the area is the
product of tiDO equal factors.
3. The factors of a number are other numbers which,
multiplied together, produce the given number.
The factors of 24 = 12-2 or
i'i
5,,
m
. 2-2-2-3
. 2"-3
4. A prime number can be exactly divided only by itself
and one.
20 MODERN JUNIOR MATHEMATICS
5. Which of the given factors of 24 are prime?
6. Find two factors of 16, 21, 23, 25, 32, 144, 49, 81, i,
A, 7.
7. Which of these numbers are prime?
8. Find the prime factors of the others.
9. Divide as many as possible into two eqiuil factors.
10. (a) We see that 24 has only two different prime fac-
tors, 2 and 3. But three 2's are multipUed to-
gether with one 3 to give 24. Thus:
24 =2. 2. 2. 3
Instead of writing out the factor 2 three times,
we may write it once and put a small 3 to the
right and a Uttle above it, to show that the 2
must be multipUed by itself 3 times. This
small 3 is called the exponent of the 2.
72 =2. 2. 2. 3-3
= 23.32
(Read, the cube of two multiplied by the square of
three or two cubed times three squared.)
(b) Write the prime factors of the following numbers
as shown in 10 (a).
(1)
(2)
(3)
(4)
(6)
12
18
20
24
25
8
27
22
28
39
9
32
14
30
42
16
35
15
36
45
48
54
63
65
75
84
85
96
100
108
125
144
150
200
225
250
175
196
300
400
11. (a) If we are given the area of a square to find
its edge, we must find the two equal factors
which give this product.
SQUARE ROOT 21
(b) This process is called finding the square root of a
number.
(c) The sign which tells us to find one of the two
equal factors is V , written over the num-
ber. It is called the radical sign.
V25 is read "the square root of 25.''
12. Given the area of a square to find its edge.
Sn =6^
25 =e2
\/25 = Ve^
V5 X 5 = y/e xe
5=6
13. 25 is called a perfect square because it is exactly 5x5.
15 is not a perfect square because there are no two like
factors which multipUed together give exactly 15.
But Vl5 = 3.87 + because 3.87 x 3.87 = 14.97 + which
is nearly 15.
Of numbers which are not perfect squares, only the
approximate square roots can be found. The degree
of approximation depends on the number of decimal
places.
Vis = 3.8 +, for (3.8)2 = 14.44
Vis = 3.87 +, for (3.87)2 = 14.9769
VI5 = 3.873 -, for (3.873)^ = 15.000129
XL Finding Square Root by Prime Factor Method
1. The square roots of numbers may bo found in throo
wajrs.
The first method of finding the square root is (lic^ \mmv
fa4ior method. It is used for finding the roots of \H»rCoct
squares.
22 MODERN JUNIOR MATHEMATICS
(a) What is the square root of 324?
Solution, a/324 = ?
V324 = VOi
= \/22.32.32
-23.3
__ no
Explanation: °
We find in 324 there are two 2's and four 3's or
two times two 3's. For each pair of equal prime
factors in the number, we take one factor for
the square root.
2. By the prime factor method find the square root
of 144, 441, 1225.
3. Find the value of V729, \/i96, \/484, V625.
4. If a square field contains 2025 sq. rd., what is
the length of each side? How many rods of fencing are
needed?
m. Finding Square Root by Mechanical Method
1. (a) The second method for finding the square root
of a number is the mechanical method.
(b) Before giving the process, let us examine the
squares of a few mmibers.
Numbers with units only have how
many digits in their squares?
Numbers with two figures or tens and
units have how many digits in their
squares?
Since 1=1^ and 100 = 10^, the square
root of a number between 1 and 100
must be a number between 1 and 10.
Since 100 = 10^ and 10,000 = 100^ the
square root of a number between
100 and 10,000 must be a number
10002 = 1,000,000 between 10 and 100.
p =
1
3« =
9
5« =
25
9" =
81
lO* =
100
15« =
225
65* =
3,025
902 =
8,100
99" =
9,801
100* =
10,000
999* =
998,001
SQUARE ROOT
23
(c) How many digits will be in the square root of a num-
ber expressed by two figures?
(d) How many figures will be in the square root of a
number expressed by three or four figures?
(e) How many digits are there in the square root of 4225?
Of 256? Begin at imits and point ofif groups or
periods of two figures each; thus 42' 25 or 42 25;
2' 56 or 2 56.
(The left-hand period may have only one figure). For
each group or period there will be one figure in the square
root.
(f) How many figures are in the square root of 3969;
441; 529; 49; 15625?
(g) On squared paper, draw a 15-inch square, using
i inch to
^ jg ^
represent an
inch.
Smce 15 = 10 + 5,
the square may be
divided as in the
figure.
How many smaller
squares does the large
square contain? how
many rectangles?
What is the area of
each?
From the figure we
see that
10" + '-5"--
+ -
ID
It
10'
X
o
10x5
5'
16* = 1(F + 2(10 X 5) + 5«
= 100+100 +25
= 225
24
MODERN JUNIOR MATHEMATICS
By multiplication we see the same truth.
15
=
10 +
5
15
=
. 10 +
5
(10
X5) +
52
10«
+ (10
X5)
15«
= 10«
+ 2(10
X5) +
5«
= 100+100
+ 25
= 225
or
tens + units
tens + units
(tens)* +
(tens X units) + (units)*
(tens X units)
(tens)* +2(tens x units) + (units)*
This statement written as a formula
is:
(t +u)* = t* +2(t xu) +u*
(h) Use the formula to find the square of 35.
Solution, 35 = 30+5
(t + u)2 = t^ + 2(t X u) + u^
(30 +5)2 = 302 + 2(30 X 5) + 52
= 900+300 +25
= 1225
Therefore 352 ^ i225
(i) Use this formula to find the square of 16, 22, 14, 17,
13, 25, 61.
(j) This study about squaring numbers will help to make
clear the mechanical method of finding square root.
2. The Mechanical Method for Finding the Square Root
of a Nimiber.
(a) Find the square root of 225; of 3969.
Explanation of V3969
Begin with imit^s digit and point off
groups of two digits each. (Why?)
The largest square in the left-hand
period is 36 or 62. Therefore 6 is the
first figure in the root.
Subtract 62 or 36 from 39, the left-
hand period, leaving 3. Bring down
the next period, making 369.
For a trial divisor (T.D.), double that
part of the root already found. That
is, double 6, the tens di©.t, which ©ves
Process
2'25 [15
T.D
= 2
1
125
CD
= 25
125
V225= 15
39'69|63
36
T.D
= 12
369
CD
= 123
369
V3^
)69 = 63
SQUARE ROOT 25
12 for a trial divisor. (Why must you double the tens
digit?) In using the trial divisor, leave off the last figure
of the dividend or 9. Try 12 into 36 or 3. Put 3 in the
root and annex it to 12, making the complete divisor (CD.)
123.
Multiply the complete divisor by the last figure in the
root. Continue this process imtil all periods are used.
(b) A square field contains 169 sq. rd. How long is the
side?
(c) Find the value of:
(1) Vm (2) Vi024 (3) Viiii (4) Vl6^
(5) V2m (6) V2209 (7) Vm (8) V^
(9) Vmi (10) V529 (11) Vl369 (12) VsiSe
(13) V4i89 (14) V656r (15) Vl5,129 (16) V46,225
(17) V3^ (18) V4356 (19) V^m (20) V6889
(21) \/ 55,225 (22) ^7056 (23) V4096 (24) V225,625
(25) V643,204
(d) Find the square roots of the following:
(1) 5184 (2) 132,496 (3) 2116 (4) 1764
(5) 2401 (6) 3249 (7) 2304 (8) 3481
(9) 9216 (10) 6241 (11) 1296 (12) 2809
(13) 6724 (14) 961 (15) 1156 (16) 123,904
(17) 9604 (18) 6561 (19) 62,500 (20) 10,609
(21) 42,025 (22) 11,881 (23) 474,721 (24) 552,049
(25) 91,204 (26) 160,801 (27) 110,889 (28) 8464
(29) 7396 (30) 402,849
3. Square Root of Decimal Fractions.
(tV)' = t*tf or (.1)^ =.01
(ThsY = Tirhm or (.01)^ = .0001
(iv^)' = T.Tnn^.Tnnj or (..001? = SRKmV
26
MODERN JUNIOR MATHEMATICS
(a) Just as with whole numbers, the square root has one
half as many decimal places as the square. So
we must point off periods of two figures, beginning
at the decimal point.
Each decimal period must have two figures. If neces-
sary, annex ciphers. .115 is the same as .1150, so the
V. 115 is the same as V.1150.
With a mixed decimal, begin at the decimal point and
point off periods to the right and left.
(b) Illustrations:
(1) Vl.5625 = 1.25
1. 56 25 1-25
1
22
56
44
245
1225
1225
(2) Vi
J = V3.OOOOOO = 1.732 +
.S. 00 00 00 1.732 +
1
27
200
189
343
1100
1029
3462
7100
6924
(c) Find t
he value of:
(1) V52.2729, V3, V2 , V.2
(^) V60.3729, V5, V.5, V.06
(3) V]
r3.96. V.025. V.016, V3.143
4. The Square Root of Common Fractions.
Therefore V^ = f
SQUARE ROOT 27
(a) If both temis of a fraction are perfect squares, to find
its square root what two roots must be found?
(b) Fmd the value of VJ, Vj, VfJ, Vff, V^.
(c) Is the square of 2 greater or less than 2?
Is the square of § greater or less than J?
(d) Is the square root of 25 greater or less than 25?
Is the square root of ^ greater or less than ^V?
(e) Find the square root of f .
We see that some other way must be found to find this
root, because 2 and 3 are not perfect squares. We may-
choose one of two ways. We may reduce such a fraction
to the decimal form and take the root, or we may make
the denominator a perfect square and solve it thus; because
= i X 2.449
= .816
(f) Another illustration:
10
22. 2*
= §x_jVio
-iVlo
= i X 3.162 +
(g) Decimal method- = '^^^^ +
f = .625
!62 50 00 00 I '7905 +
49
149
15,805
1350
1341
90,000
79,025
Note: The value of the square roots of such numbers as 2, 3, 5, 6,,
7, 10 . . . are kept in convenient tables by those who use them a great
deal. For such persons the method given in (.f) is shorter than the
decimal method.
28 MODERN JUNIOR MATHEMATICS
(h) Find the value of VI, Vj, Vj, >/|, VI Vf, Vf, V|.
6. Changing Common Fractions to Decimal Forms.
(a) We learned that, to find the square root of most frac-
tions, they must be reduced to decimal form. By division, we
discovered that some fractions can be changed to decimal
form exactly or without remainder. Such fractions are called
terminating decimals (decimals with an end). Others do
not terminate, no matter how far one carries the division.
(b) By inspection, one can tell whether or not the frac-
tion is terminating, and thus save the time of carrying out
lengthy ^division in the hope that it will finally come out
"even" or without remainder. Decimal fractions are those
having ten or a power of ten as their denominators; as
iV> ihsi ToW> loioo ' These may be written as
_L _L. J_ _J_
2-5' 22.52' 23.53' 2*54
We see, therefore, that every decimal must have only 2's
and 5^s as factors in the denominator, and that the niunber
of 2's and 5's must be the same.
(c) If any fraction in its lowest terms has any factor other
than 2 or 5 in its denominator, it can never be reduced to a
5 5
terminating decimal; for example, 77^ = 7^i~^. Since one catt-
le J • o
not get rid of the factor 3 in the denominator, one should
know by inspection that it can never be changed into a
terminating decimal. The division should be carried to as
many decimal places as needed, perhaps three or four.
A = .4166 +
.4166 +
12 5.0000
(d) Every fraction, which in its lowest terms has only 2's
and S's in its denominator, may be easily elvan^ed to
SQUARE ROOT 29
decimal form by making the nimiber of 2's and 5's equal.
Thus
1=^=^-X^' = ^^= 075
40 5-2» 5 •23'^ 52 53-23
Explanation:
3
Factor the denominator 40 and write the fraction as
5-2»
Since there are three 2*8 and only one 5, we must multiply both
75
terms by 5^ which gives
5»-2»
5" • 2* = 10* or 1000, therefore three decimal places are necessary. The
decimal form is always the numerator and the number of decimal places
is the number of factors 2 or 5 in the denominator. In other words,
the number of decimal places equals the exponent of the factors of the
denominator.
(e) Other examples:
1 =. J _ 3 '5^
16 2* 2*. 5*
7 7 7.53
80 2*. 5 2*.5-53
= .1875
= .0875
(f) Tell by inspection which of the following f ractioAS can
be changed to terminating decimals. Change them by the
factor method.
2 9 If 4 J TSf Tifj TSf S) TSy Tff> ITlfJ %> 9
IV. Finding Square Root by Inspection
The third method of finding square root is by in-
spection. Its use necessitates one's knowing the squares
of all numbers from 1 to 30. The method is exglainad \3x
the supplement
30 MODERN JUNIOR MATHEMATICS
V. Review Problems
1. Find the square root of:
(a) 4761 (e) 1960
(b) 585,225 (f) 2250
(c) 4624 (g) 5335
(d) 6255 (h) 256,036
2. Find the value of the following:
(a) vl_ (c) VA (e) vS fe) VA
(b) VtV (d) VA (f) VA (^) ViAr
CHAPTER THREE
A STUDY OF TRIANGLES
I. Reading of Angles
1. Cut out a square and carefully fold it between two
opposite comers. i> c
2. The name of the line made by
the fold is diagonal.
Diagonal comes from two Greek
words which mean through the angle.
3. Into what kind of figures is the
square divided by the diagonal?
4. The word triangle means
three-cornered. We write it thus, A;
plural A.
6. (a) Do the two parts of the square exactly fit each
other?
(b) What part of the area of the square is in the tri-
angle?
(c) If the side of the square is 4, what is its area?
(d) What is the area of each triangle?
<6. (a) How does the diagonal divide the angles at the
comers of the square at A and C?
(b) How big is each angle of the
triangle?
7. This figure may be named the
triangle ABC, (A ABC).
Each comer is the vertex of the angle
made by the lines that meet there.
8. How to read the names of angles,
(a) Sometimes we name an angle
with three letters, reading from one end of a
side to the vertex, then to the other side.
31
32 MODERN JUNIOR MATHEMATICS
(1) In the triangle, the Z at the vertex A is read
Z BAC,
In the triangle, the Z at the vertex B is read
Z ABC
In the triangle, the Z at the vertex C is read
AACB.
B
(2) In the name of the angle, where is the vertex
letter always found?
(b) Sometimes a small letter or number is placed
within the angle close to the vertex; as, Zx,
Zl, Z2.
9. The size of the angle depends upon the opening be-
tween the sides and not upon the length of the lines.
The Z.A may be read
Z.BAX, ZBAYy ZBAZ
ZCAX, ZCAY, ZCAZ
ZDAX, ZDAY, ZDAZ
D
Using different points on the
sides of the angle does not
change its size.
XL The Sum of the Angles of a Triangle
1. Since ZB is an angle of a square
ZB = 90° or art. Z.
2. ZA:^ ZC ^ 45° = I rt. Z. Why?
TRIANGLES FROM SQUARES
3. What is the sum of AA^ B, and C?
ZB= 90° = Irt. Z.
45° = i rt. Z .
45° = irt. ^.
Zil =
ZC =
ZJ5 + Zil + ZC = 180° = 2 rt. A.
4. Cut off the three comers of the
triangle and place them carefully about
the point of your folded Z paper, thus:
G
What does this show?
6. If the three comers are tinted with different colored
crayons, the parts may be seen more distinctly.
6. Cut out many differently shaped triangles and test
the sums of the angles in the same way.
7. Use your protractor to measure the three angles of
several triangles and see if their sum is 180°.
8. What is your conclusion as to the sum of the angles
of any triangle?
34
MODERN JUNIOR MATHEMATICS
(a) Draw AASC.
(b) Draw a perpendicular from the vertex of the
largest angle.
(c) Cut out the triangle and fold over the comers so
as to meet at the foot of the perpendicular.
(d) Wbat does this prove?
m. Complementary Angles
1. What is the sum of the two small angles of a right
triangle?
2. Any two angles whose sum is a right angle or 90° are
complemeniary angles. Each angle is the complement of the
other.
C
ZA-n-
3. Angles A and B are complementary angles because
their sum is the rt. /.COD,
4. If ZA = 65°, then ZB = 90° — 65° = 25°.
If ZA = n degrees, how many degrees are in ZB?
6, With your protractor draw the following angles.
Compute and draw the complement oi eack.
(a)
30°
(f) 68°
(k) 19°
(p) h rt.
(b)
23°
(g) 5°
(1) 35°
(q) ^ rt.
(c)
45°
(h) 85°
(m) 80°
(r) |rt.
(d)
75°
(i) 60°
(n) 6°
(s) f rt.
(e)
12|°
(j) 2h°
(o) 90°
(t) Jrt.
TRIANGLES FROM SQUARES 35
Z.
Z.
Z.
Z.
Z.
IV. Interior and Exterior Angles of Polygons
1. We call angles within a triangle or parallelogram, in-
terior angles to distinguish them from exterior angles which
are otdside the figure.
2. Exterior angles of triangles.
(a) To form an exterior angle of a triangle or other
polygon extend any side in one direction only.
The angle formed by one side of the triangle
and an adjacent side extended is an exterior
angle.
(b) A Xy y, and Z are all exterior A.
Can you make others in these triangles?
(c) Draw a figure as A ABC. Cut out the A Sit A
and C and carefully fit them into the exterior
ZX. Do this with several triangles, or imtil
you feel sure that:
An exterior angle at one vertex of a triangle is
exactly equal to sum of the two interior angles
at the other vertices.
(d) Test this by measuring the A m\Xi ^«v3;x t^x^
tractor.
36
MODERN JUNIOR MATHEMATICS
B. TRIANGLES FROM RECTANGLES
1. You have found that the diagonal of a square divides
it into two equal triangles.
2. Draw and cut out a rectangle. Fold on one diagonal.
Do the two parts fit?
3. Cut along the diagonal. Now can you make the
two parts fit?
4. What part of the rectangle is now in each A ?
D C
A B
6. (a) In our first formula for the area of a rectangle we
used the names length and width because they
were the names for the dimensions of the oblong
block. We have other names. The side on
which the rectangle stands is the base, as the line
AB. The side perpendicular to the base is called
the height or altitude, as the lines BC or AD.
(b) We foimd that the area of
the surface of a rectangle = length x width, or,
the surface of a rectangle = base x height.
Therefore we may write the formula thus:
Sn = 6/1
CLASSIFICATION OF TRIANGLES
37
C. CLASSinCATlON OF TRIANGLES
L According to Size of Angles
Triangles have different names according to the size of
their angles.
1. A triangle having one angle a right angle is a right
triangle.
2. A triangle having one angle an obtuse angle is an o6-
tjise triangle.
3. A triangle having all angles acute is an acute triangle.
An acute triangle having all of its angles equal is called
an equiangvlar triangle. Why?
Right
Obtuse
Acute
Equiangular
IL According to Equality of Sides
Triangles have different names also according to the
number of the sides that are equal.
1. A triangle having its 3 sides equal is an equilateral
triangle. Equilateral means eqv^al sided.
Equilateral
Isosceles
Scalene
2. A triangle having 2 of its sides equal is an isosceles
triangle. Isosceles means eqiial legs.
3. A triangle having no sides equal is a scalene triangle.
Scalene means uneven.
4. We seldom use the word scalene; but when we mean
either of the two special kinds, we are very careful to specify
them by name.
38 MODERN JUNIOR MATHEMATICS
6. (a) Draw one or more of each kind of triangles as
accurately as possible.
(b) Label each kind.
(c) Measure the angles in each.
(d) Find the sum of the angles in each triangle.
D. AREAS OF TRIANGLES
I. Area of Right Triangle
1. We have seen that the triangles formed by the
diagonal of a rectangle are two equal right triangles.
2. Those formed by the diagonal of a square are two
equal isosceles right triangles.
3. Since S □ = 6/i
and since the rt. A == | D
therefore, Sit. A = {bh
4. Which sides of the right A may be the base and
height?
6. The two ± sides are called the legs of the rt. A.
6. The longest side, the one opposite the right angle, is
called the hypotenuse,
7. Hypotenuse means stretched under. Which angle is it
stretched under?
8. The hypotenuse may be considered the base of a right
triangle. In that case, the altitude or height is a line drawn
from the vertex of the rt. angle ±
^^x.,^ the hypotenuse.
^\^ 9. How to find the altitude
^\^^ of any triangle (except certain
-^ altitudes of obtuse A),
(a) By folding.
(1) Draw and cut out a triangle. Make a fold
from one vertex to ttie op\>o^\\.^ ^ida, ^o \Jaa.\>
AREAS OF TRIANGLES 39
the two parts of that side fall exactly to-
gether.
Unfold and measure the A on either side of
the fold with your protractor or /-paper.
This fold is one of the altitudes of the A-
(2) Make a similar fold from each of
the other vertices.
(3) Any side may be the base of a A ,
but with each base goes the X
to it from the opposite vertex.
(4) Do the three altitudes you have
folded meet in the same point?
If not, try again, for they should
meet.
(b) By using the protractor.
Shde the end of your protractor
,refully along the base of your
■iaugle until one side just
reaches the opposite vertex.
Then draw the alti-
tude.
(c) More exact constructions with a compass will be
shown in Chapter Four,
n. Area of Scalene Triai^le.
1. (a) On squared paper, draw a rectangle whose base
is 5 inches and whose height is 4 inches. Plain
paper may be used, but you must be vevv *iMe^
Aii to make exact right ang^ea.
40 MODERN JUNIOR MATHEMATICS
(b) At E, any point in the base not the mid point,
draw EF ± AB. Use protractor or folded
paper to make the rt. Z if plain paper is used.
(c) Draw the diagonals AF and BF.
(d) ABCD is a □, with base AB and height EF.
(e) ABF is a A, with base AB and height EF.
(f) Cut out the □ ABCD and cut along AF and BF.
(g) Fit A ADF on A AEF and
fit A BCF on A BEF.
(h) We see then that A ABF = | □ ABCD.
But, S O ABCD » bh = 5x^ = 20 sq. in.
Therefore S A ABF = ^ M ^^^ | x 5 x 4 = 10 sq. in.
J AREAS OF TRIANGLES 41
(i) Since ABF is a scalene A, the formula,
S A = J &/i, must be true for any triangle.
Translate this formula into an EngUsh statement.
(j) Check the area by counting the small squares or
equal parts of squares on the squared paper.
(k) Test this formula with other figures.
2. Exercises.
(a) Draw three altitudes in a triangle. Measure
each base and altitude to the nearest tenth of
a centimeter. Find the area from the three sets
of measurements. Are they approximately
equal?
(b) Find the area of a triangle whose base is 6' 4" and
whose height is 4' 6".
(c) Find the area of the following triangles, using the
formula.
Estimate the areas first:
h
h
S
(1)
9"
6"
(2)
4!"
2i"
(3)
5' 8"
3' 4"
(4)
6.6'
3.5'
(5)
120 rd.
62.5 rds.
(6)
17i"
13J"
(7)
3' 9"
2' 3"
(8)
2'
18"
42 MODERN JUNIOR MATHEMATICS
(d) A farmer has an irregularly shaped field ABCD.
By dividing it into two triangles, he can meas-
ure each to find the area of the field.
What three Unes must he measure?
If BD = 280 feet, CX = 75 feet, and 47 = 190
feet, find the area of the field.
CHAPTER FOUR
CONSTRUCTIONS
A. CONSTRUCTIONS OF TRIANGLES
1. On squared paper it is easy to draw squares, rec-
tangles, and other figures correctly, but with a ruler and
compass we can draw exact figures of all kinds on plain paper.
Before experimenting any further with measurements of
different figures we shall learn to construct them with our
instruments.
Some suggestions follow:
(a) Always keep pencils well sharpened. A pencil sharpener should
be in every mathematics room.
(b) Be very exact in all measurements.
(c) Keep work neat.
(d) One cannot be too painstaking in all construction work.
(e) Keep all figures of reasonable size, neither too large nor too small.
(f ) Do not lift compass point any more than necessary.
2. To construct a right C
triangle,
(a) Draw a line AB any
length.
With your protractor
draw a line AC ±
AB at point A.
Make AC any A'~ B
length. Draw BC, Then A ABC is a rt. A.
(b) With your compass draw a circle.
(The si^ for ch-cle is O.)
Through the center 0, made by the point of the
compass, draw a line cutting the circle in two
parts.
43
44 MODERN JUNIOR MATHEMATICS
(This line is a diameter. It means the measure thrcmgh,
that is, through the center.)
Letter this diameter XF. Take
any other ipoint on the O as Z,
and draw XZ and YZ. With
your protractor measure the ZZ.
1^ ^-^, ^^Y What kind of a A is XYZ f Why?
Note: The ZZ is said to be
inscribed in a semicircle, because its
vertex lies on the circle and its sides
pass through the ends of the diameter.
The statement that "every angle that is inscribed in a
semicircle is a right angle" was proved to be true by a fa-
mous Greek over 2500 years ago. The man was Thales,
who was born about 640 B.C.
At the time Thales Uved, there was no such thing as
arithmetic or algebra, and only a very few facts of geometry
were known. His discovery of the truth about the right
angle being inscribed in a semicircle was the cause of a great
celebration, and it is said that he sacrificed an ox to the im-
mortal gods.
When you study geometry in the senior high school, you
may learn how Thales proved this
theorem.
3. To construct an acute
triangle,
(a) With your protractor,
draw a triangle with
angles less than 90°.
(b) In a circle, construct an
acute triangle as follows :
Draw a circle and a line
not a diameter, as MN.
Draw a diameter MX. On the upper part of the G
between M and X take soinft pavxv\, w^wc ^tl<5\\^
CONSTRUCTIONS OF TRIANGLES
45
to X so that when PM and PN are drawn, the
. center will be within
the A.
With your protractor meas-
ure all the A of the
A PMN. What kind of
A is it? Why?
4. To construct an obtiLse/\^
triangle.
(a) Draw a O and the line MN
. as in exercise 3 . Take P
any point in the lower part of the O. Draw PM
and PN. With your protractor measure ZP.
What kind of A is ilfiVP? Why?
(b) With your protractor make an obtuse Z at any point
A. Draw a line cutting both sides oi ZA. What
kind of A is ABC? Why?
5. (a) The curved line which
makes a circle is
sometimes called its
circumference. Any
small part of a drcum-
ference is called an arc. We write it thus, r^j r^.
(b) The point where two lines meet or cut each other
is called their point
of intersection or their
intersection. Intersect
means to cut into.
We speak of two
intersecting streets,
meaning two streets
that cross each other.
6. To construct an equilateral
triangle.
(a) Draw a line XY as \on^ as yow v^^w\> ^^ '^^^ ^
46 MODERN JUNIOR MATHEMATICS
your A. Measure the line XY between the
points of your compass.
With the sharp point on X make an arc above
the Une. Then put the sharp point on Y, and
without changing your compass make an arc
that will intersect the first one, as at Z. Draw
XZ and YZ.
(b) With your ruler measure the length of the three
sides of the A. Measure them with your
compass. With which one can you measure
more exactly? What kind of A is XYZf
Why?
With your protractor measure the three A,
What other name can you give? How many
degrees are in each Z of an equilateral A?
7. To construct an isosceles triangle.
(a) Draw a line MN as long as you want the side of
your A. With your compass measuring more
than MN draw arcs as before, intersecting at 0,
Draw OM and ON as in Fig. A.
In Fig. B draw the arcs with a measure less than
MN. What kind of A are MON and MPNf
(b) Is every equilateral A isosceles? Is every isos-
celes A equilateral?
(c) With your protractor measure the A M and N.
Make a nmnber of isosceles angles and measure
the angles at each end of the base. These
angles are called base angles.
CONSTRUCTION OF LINES 47
(d) What have you discovered about the two base A
of an isosceles A?
(e) The other angle, i.e. the angle opposite the base,
is called the vertex angle,
8. To construct a scalene triangle.
yn
rt
-X
Draw three lines of different lengths, as m, n, o. On any-
other line, as AX, with your compass measure AB = m.
With yoiu- compass measure n, and with the sharp point at
A make an arc.
Likewise with the measure o and the point at B make
an arc intersecting the other arc at C. Draw AC and BC.
What kind of triangle is ABC? Why?
B. CONSTRUCTION OF LINES
L To Draw Perpendicular Lines
1. To draw a line perpendicular to another line at a given
point in the line,
(a) By using a protractor.
Suppose P is the point in the given line XY at
which a ± is to be erected.
Place the protractor so that the 90° angle line
coincides with the line XY. Slide the pro-
tractor until its edge just touches point P,
being careful to keep the two lines exactly to-
gether. Draw a line through P along the
edge of the protractor, as OP. T\v^ \xsi^ O^
is perpendicular to tlie Ame XY ^\» P .
48
MODERN JUNIOR MATHEMATICS
For yourself find another way of using your
protractor to draw a perpendicular to a given
line at a given point.
O
X
(b) By using a compass.
Suppose P is the point
in the Une AB at which
the ± is to be erected.
Place the sharp point of
the compass at P and
draw a semicircle (half
M P N B circle) which cuts iljB at
M and iV, or draw two arcs cutting AB at M
and N, Open the compass a little more, put
the sharp point first at M and draw an arc
above P, and then at N and draw an arc inter-
secting the first one at 0. T)xa>N ^ Xvtv^ l\««\.
CONSTRUCTION OF LINES
49
to P. OP is ± to AB. Measure the ZAPO
and ZBPO with your protractor.
What does the size of these A prove about the
line OP?
2. To draw a line perpendicular to another line from a
joint not in the line, p
(a) Suppose P is
the point
from which
a ± is to be M u ^
^
B
>tQ
drawn to
MN. Place
the sharp
point of the
compass on
P, and with an opening great enough, draw
two arcs cutting the line MN in two points, as
A and B,
Put the sharp point of the compass on A and draw
an arc below P, then on B, and draw an arc
cutting the first at Q, Draw a line from P to Q.
Call the point at which PQ cuts MN the point 0.
(b) Measure the A NOP and MOP.
What do these measures show?
(c) From P draw several lines to MN,
p PAi is read PA sub 1.
PA 2 is read PA
sub 2. The same
kind of lettering
with different num-
bers written below
the letters shows that the lines were drawn in
the same way but at different points. How are
all the lines PA, PAi, PA^, P^^^ ^t?l^\v>. kx^
these lines longer or shortei tYv^xi \o\vei 1. P01
50
MODERN JUNIOR MATHEMATICS
(d) Measure the angle that each of these lines makes
with MN. How do they compare with Z PONf
As the point A moves away from 0, what happens
to the size of the Z ?
How many lines can you draw from P ± to MNf
(e) What is the shortest line that you can draw from
a point to a given line?
(f ) The distance from a point to a line always means
the shortest distance.
(g) What is the distance of P from MN ?
(h) Do you beUeve the following?
(1) The shortest distance from a point to a line
is the perpendicular drawn from the point
to the Une.
(2) One and only one perpendicular can be drawn
to a line at the same point.
(3) If you beUeve the above statement No. (2), ex-
amine your cube or oblong.
How many lines are _L to DB at point Bf
Can there be more than two lines Jl to
DB at Bf
How many can there be, if we consider
only one face or plane surface at a time?
Would statement No. (2) be correct if we
limit it to lines in the same plane?
In the same plane, one and only one perpen-
dicular can be drawn to a line at the same
point,
(4) Is this statement true?
In the same plane, one and only one perpen-
dicular can be drawn from a given point to
a line.
CONSTRUCTION OF LINES
51
1.
n. To Draw Lines of Bisection
To hised a line.
c
M
B
D
(a) Suppose AB is the line to be bisected (cut in
two equal parts).
Open the compass a little more than half of the
line AB. Put the sharp point on A and draw
one arc above and another below the line.
Then from B make two arcs intersecting the
others, being careful not to change the
opening of the compass. Call the points
of intersection C and D, and draw a lino
between them, which cuts AB at some ix)int,
as M. Then M is the midpoint of AB, or
AB is bisected at M.
(b) With compass show that AM ^ MB.
(c) With compass and protractor show that CD is
the perpendicular bisector of AB,
2. To bisect an angle.
(a) By folding.
You have already folded a papor twioo to make
rt. A. Fold such a papor av;aiu« Wlrni \s»v tXvi.
size of the new auftW? ¥oV\ i\vkvv\\\. "Wcv^V Ss.
52
MODERN JUNIOR MATHEMATICS
the size now? Cut out an angle of any size and
bisect it by folding,
(b) By using a compass.
Suppose LAOB is to
be bisected. Put the
compass point on
and draw two arcs
cutting the sides of
the angle at X and Y,
Put the compass point
on X and then on Y
and, without changing the opening, draw two arcs
intersecting at P. Draw a line from through
P. What does OP do to /.A0B1 Cut out
the /.AOB and show by folding that it is bi-
sected.
CHAPTER FIVE
FURTHER STUDY OF TRIANGLES
A. IMPORTANT LINES IN TRIANGLES
L la Isosceles Triangles
1. Draw and cut out a number of isosceles triangles of
different sizes and shapes,
(a) Fold each through the vertex angle so that the
two edges he one on the other.
(b) Do the base angles exactly fit?
(c) Does this test verify the measure with your pro-
tractor?
(d) Unfold one or more of these triangles.
Measure the two parts of the base.
In what two ways can you show that the fold bi-
sects the base?
(e) What kind of angles does the fold make as it
meets the base?
In two ways prove your answer correct.
What name do you give to a line that makes
right angles with another Une?
(f) With your protractor measure the two parts of
the angle at the vertex.
When the isosceles triangle is folded, do these two
angles exactly fit?
What else besides the base of the triangle is bi-
sected by the fold?
2. Draw three isosceles triangles exactly equal.
(a) In A -4, bisect the vertex Z.
What does the bisector of the vertex Z of an isoace-
les A do to the base? How doeaV\iTciRfc\»\5w$i\j«i^-
53
64
MODERN JUNIOR MATHEMATICS
(b) In A B, draw the _L bisector of the base.
Extend this _L bisector. What point of the A
does it pass through?
L B c
What does the perpendicular bisector of the base
do to the angle at the vertex? Test carefully.
(c) In A C, draw a J. from the vertex to the base.
What does this ± to the base do to the base?
What does it do to the Z at the vertex? Test
with compass and protractor.
n. In Scalene Triangles
1. Draw and cut (mi three
scalene triangles,
(a) In one of these triangles
make three folds.
(1) An altitude
(2) A [bisector of the
vertex angle
(3) A perpendicular bi-
sector of the base
(4) Are these folds the
same or are they
three separate
IMPORTANT LINES IN TRIANGLES
55
(b) In the second triangle, very carefully bisect by
folding the three angles of the triangle.
(1) Do these three bisectors meet in a point?
(2) Is it the same point in which the three alti-
tudes meet?
(3) Draw a triangle and accurately construct the
bisectors of the angles. Call their inter-
section 0.
From draw a ±
to the base.
With this ± as
a radius and
as a center draw
a circle. Does
your circle just
touch the three
sides of the tri-
angle? If drawn
very carefully,
it will.
(4) Such a circle is
inscribed in the triangle, i.e. drawn inside
the triangle.
'How did you find its center?
How did you find its radius?
(6) Inscribe a circle in an equilateral triangle.
(c) In the third triangle, fold carefully the ± bisectors
of the three sides.
(1) Do these bisectors meet in a point?
(2) Is it the same point as the other intersections?
(3) Draw a triangle and accurately construct
the ± bisectors of the three sides.
Call the point of intersection E.
56
MODERN JUNIOR MATHEMATICS
Draw a line from E to one of the vertices of
the triangle.
With this line as a radius and £ as a center,
draw a circle.
Does your circle pass through the other two
vertices? If drawn carefully, it will.
(4) Such a circle is called a circumscribed circle,
i.e., it is drawn about the triangle.
HL Designs
(1) Circumscribe a circle about an equilateral
triangle.
(2) Circmnscribe a circle about an equilateral tri-
angle, and inscribe a circle in the same
triangle.
(3) Draw an equilateral triangle.
With each vertex as a center and a radius
equal to one-half of a side, draw arcs that
meet at the midpoints of the sides. Such a
figure is called a trefoil and is used exten-
sively in architecture.
IMPORTANT LINES IN TRIANGLES
(4) Draw the following patterns or similar ones.
Vary them with different shadings or colors.
One similar to Fig. B may be made by divid-
ing the diameter in five equal parts instead
of three.
58 MODERN JUNIOR MATHEMATICS
IMPORTANT LINES IN TRIANGLBB 59
mm
60
MODERN JUNIOR MATHEMATICS
B. THE RIGHT TRIANGLE
I. Its Properties and Practical Uses
The right triangle is a very mteresting figure because of
its peculiar properties and its practical uses.
In making trusses for bridges and buildings, the steel
beams are put together in the form of triangles for the sake
of security, for a triangle cannot change its shape as long as
the lengths of its sides do not change.
Four rods hinged together may be a rectangle, but pressure
or strain may change it to a parallelogram without breaking
or changing its sides. Show how this fact is used in tele-
phone brackets on office desks.
/7^
Such a change is evidently impossible in a triangle.
Fig. A shows a kind of truss sometimes used when the
strain is not to be very great; but Fig. B shows one of
many ways of strengthening it.
If the strain is to be exceedingly great, the second diagonal
beam is placed in each rectangle.
The right triangle is a necessary tool for nearly every
artisan and craftsmaji^
THE EIGHT TRIANGLE
^<
^
^
i
P'
^
F"
\
,'•'
/•'
■.
1. (a) On a piece of squared paper draw alternate
diagonals and shade a square on each side of
the right triangle ABC. What is the aide AB
called?
(b) How many small triangles are in the square on the
hypotenuse?
(c) How many are in the square on each leg?
(d) How does the size of the square on the hypotenuse
compare with the sum of the squares on the
other two sides?
(e) The A ABC is a special right triangle.
Which kind?
2. (a) Draw a triangle with sides 3, 4, and 5 centimeters
long. (If more convenient a half inch may be
used for a centimeter.)
(b) What kind of triangle is formed?
(c) Carefully construct a square on each side and
divide each carefully into sqaate (;is&'C\m'*KK.
MODERN JUNIOR MATHEMATICS
(d) How does the square on the hypotenuse compare
with the sum of the squares on the other two
sides?
{The line BC squared
equals the line AB squared
BC ~AB + AC
25 = 16 + 9
25 == 25
plus the line AC squared.
Or, the square on BC
equals the square on AB
plus the square on AC.)
(e) The truth we have found for two right triangles
is proved in geometry for all right triangles.
The statement is called a theorem and is the
most famous of many in geometry. It was dis-
covered by an illustrious Greek named Pythag-
oras, who lived over 500 years before Christ
(569 B.C. to 501 B.C.). It is called the Pytha-
gorean theorem. Over one hundred different
THE RIGHT TRIANGLE 63
proofs of this theorem have been discovered,
one of which was originated by the late Presi-
dent Garfield when he was a boy.
Three thousand years before Pythagoras Uved,
the Egyptians knew that a triangle whose sides
were 3, 4, and 5 units long was a right triangle.
The ancients used this theorem in laying out their
temples which had to face a definite direction.
They tied twelve knots in a rope of long grass
or reeds at equal distances apart, and placed
three stakes in such a position that the rope of
grass would just reach around them, with one
stake at the third knot and one at the seventh.
The men who found the directions in this way
were known as "rope-stretchers."
(f) We have seen that a 3-4-5-sided triangle is a
right triangle. Would a 6-8-10-sided one be a
right triangle also?
(g) Find out how a carpenter or builder uses a 10-foot
pole to be sure that the comers of his house are
"square," i.e., right angular.
(h) A statement that two numbers or quantities are
equal is called an equation.
Formulas are usually written as equations; as,
12 =4.3
(i) From the Pythagorean
theorem we have the
equation
c2 =: a2 + 62
A
if c is the hypotenuse,
and a and h are the legs of a right triangle.
64 MODERN JUNIOR MATHEMATICS
(j) We know that i = .25.
Extract the square root of each side of the equation.
i = .5
Are the square roots equal?
If we extract the square root of both sides of an
equation or formula, we still have an equation.
(k) In the rt. triangle, (? ^ a^ + 1^
Therefore c = Va* -h 6*
If a = 3
and 6 = 4, c = Vs* + 4^
= V9 -hl6
= V25
= 5
3. Find the hypotenuse .of the right triangles whose
perpendicular sides are as follows:
a
6
c
(a)
5"
12"
(b)
45'
60'
(c)
1.5"
2"
(d)
15'
36'
(e)
8"
15"
(f)
2' 6"
6'
(g)
4'
7' 6"
(h)
8"
10"
(i)
7"
24"
0)
2"
3"
(k)
3' 6"
12'
0)
10'
12'
•
(m) Find the length of the diagonal of a square whose
side is 5
//.
'//.
2V'
THE RIGHT TRIANGLE
65
(n) A telegraph pole is 30' high. How long a guy-
wire will be needed to fasten it to a stake 16'
from tlie foot of the pole, if one foot is allowed at
each end for fastening?
(o) Suggestions,
(1) In problem 3 (b) you were to find the hypot-
enuse of a right triangle whose sides were
45 and 60.
Did you recognize 45 as 3 x 15?
and 60 as 4 x 15?
Therefore the hypotenuse is 5 x 15 or 75.
(2) In 3 (a) you found another groUp, 5, 12, and
13, that makes a right triangle.
(3) How many problems in the set are based on
others already found?
(4) Only six of these twelve problems need to be
worked out in full. The others may be
solved by inspection.
Is it not worth while to try to factor the num-
bers first?
4. (a) Draw an isosceles right triangle with legs 1 inch
long, and draw the squares on the three sides.
66 MODERN JUNIOR MATHEMATICS
(b) According to the Pythagorean theorem how does
the square on c compare with o*? With Vf
How many square inches are in c*?
(c) Measure the length of c with your ruler. Can
you measure it exactly?
If c2 = 2
then c = V2
(d) On a straight line carefully marked oflf in inches
and tenths of inches, lay oflf the exact length of
the hypotenuse c.
12 3
'«''' I '«'» i «>''>«'»» I '«'«>'«'' I
I I I
We see that the mark comes between 1.4 and 1.5
inches. If we could divide our tenths of an
inch into tiny hundredths of an inch, the mark
would come between 1.41 and 1.42 inches.
(e) Find the square root of 2 correct to four decimal
places.
v^ = 1.4 H- (1.4 -h)2 « 1.96 +
V2 = 1.41 + (1.41 +y = 1.9881 +
V2 = 1.414 + (1.414 +)2 « 1.999396 +
V2 = 1.4142 -f (1.4142 +)' = 1.999961 +
We see that V2 is a real and exact measure of the
hypotenuse, while the measures on the scale
Une are only approximate, but growing more
nearly correct as we add decimal places.
5. (a) Draw a right triangle with a base equal to c or y/2
and an altitude of 1 inch.
a2 = ?
62 = ?
C2 = ?
c =?
THE RIGHT TRIANGLE
67
(b) On the scale lay off the length of the hypotenuse
c. Between what two marks on the scale does
it he?
(c) Find the square root of 3 correct to four decimal
places, and find the squares of the successive
values.
6. (a) Draw a right triangle whose legs are 1 and 2
inches, respectively. What is the area of the
square on the hypotenuse?
(b) What number represents the exact length of the
hypotenuse? Find its approximate value on the
scale.
7. (a) What are the lengths of the sides of a right tri-
angle whose hypotenuse is exactly Vg?
(b) Such numbers as V2, Vs, Vs, Vg, etc., whose
values can be represented exactly by lines, but
only approximately by ordinary numbers, are
sometimes called irrational numbers.
\^ is just as real as 1.7S205, andLSaxctfyt^ ^-^a^^X*.
68 MODERN JUNIOR MATHEMATICS
Many times it is more convenient to use the ir-
rational form instead of the ordmary form.
(c) How would you draw a square whose area is ex-
actly 3 square inches? 4 sq. in.? 5 sq. in.?
6 sq. in.? 10 sq. in.?
CHAPTER SIX
PARALLEL LINES
A. MEANING OF PARALLEL LINES
1. What does parallel mean?
Explain what is meant by parallel lines.
2. Is it possible that two lines could never meet and yet
not be parallel?
3. Draw a diagonal in the upper face of a cube. Can
such a diagonal be prolonged far enough to meet any lower
edge?
Are they parallel?
4. How must the position of two parallel lines be limited
in your definition?
B. CONSTRUCTION OF PARALLEL LINES
I. With Protractor
1, To draw parallels with protractor.
Draw three or more Unes JL to XY,
What kind of lines have you drawn?
— B
D
£
69
70
MODERN JUNIOR MATHEMATICS
B
n. With T Square
I. To draw parallels with T square.
Draftsmen and carpenters draw parallel lines thus with
T squares.
Suppose a carpenter wishes to cut off 4-inch strips from a
12-inch board. First he
"squares off" the end by
placing the T square in
position A. Then across
the true end he notes the
4-inch and 8-inch marks,
places his square as in
position B and draws his
lines for sawing.
m. With Drawing Triangle
1. To draw parallels with drawing triangle.
For finer work, draftsmen use a right triangle, usually
made of celluloid.
(Pupils should have two right triangles, one with the
acute angles 30° and 60°, the other an isosceles right triangle.
These may be made of cardboard.)
• ^iven point
CONSTRUCTION OF PARALLEL LINES 71
(a) AB is the given line and P a point not on AB^
through which a Hne is to be drawn || to AB.
(b) Place the drawing triangle with any edge, prefer-
ably the hypotenuse, exactly fitting the given
line. Place your ruler so as to fit one of the
other edges of the drawing triangle.
(c) Hold the ruler still and sUde the triangle along it,
keeping the edge carefully fitted against the
ruler, until the hypotenuse passes through P.
(d) Draw a line CD along the hypotenuse passing
through P. Then CD is || to AB.
Note: With ruler and drawing A using the legs of the A, i.e. the
_L sides, show how one can draw a J. to a given line from a given
point.
IV. With Ruler and Compass
1. To draw paraUek with nder and compass.
Let P be the point through which a line is to be drawn
II to AB.
(a) Through P draw any line meeting AB at 0.
(b) At P, draw an Z MPN = Z POB.
(c) The Une PN is jj to AB.
2. Second method with ruler and compass.
This method is given in a Freweiv book ^\&>te^Vsfc4 xsv
1728.
72
MODERN JUNIOR MATHEMATICS
C
B
B
Let C be the point through which a Une is to be drawn
to AB.
(a) Draw any Une from C to ABj as BC.
(b) With BC as a radius and B and C as centers, draw
the ^ AC and BD.
(c) Measure the ^ AC and make ^ BD = ^^ AC.
(d) The Une CD is || to AB,
(e) Fig. B shows the same construction with only
parts of the Unes drawn.
C. PARALLELS CUT BY A TRANSVERSAL
A
h
C
\
\
J
9
T
• \ •
tV]
w
Draw several sets of two or more parallels, with a Une
drawn across or cutting the parallels. Such a Une is called
a transversal (meaning turned across).
If two paraUels are cut by a transversal, eight angles are
formed. For convenience we give these angles different
names, depending upon their posVtioxv.
PARALLELS CUT BY A TRANSVERSAL 73
L Angles Made by a Transversal
1. The angles lying between the parallels are called in-
terior angles (mside the parallels), as angles 3, 4, 5, and 6.
2. The angles lying wUhoiU the parallels are called ex-
terior angles (outside the parallels), as angles 1, 2, 7, and 8.
3. (a) The angles lying on opposite or aUemate sides of
the transversal are called aUemaie angles, as
angles 3 and 6 or angles 1 and 8.
(b) If angles 3 and 6 are a pair of alternate interior
angles, what may you call angles 1 and 8?
(c) Select and measure the alternate interior angles.
How do they compare?
4. The angles lying on the same side of the transversal
and on the same or corresponding sides of the parallels are
called corresponding angles, as angles 2 and 6, or angles 3
and 7.
6. The fourth method of constructing two parallel lines
was by making two equal corresponding angles.
6. Select and measure all the corresponding angles in
these figures.
What do you conclude about their equaUty?
n. Vertical Angles
1. The angles lying opposite each other at the same
vertex are called vertical angles, as angles 1 and 4, or angles
2 and 3.
2. If any two straight lines intersect, how many pairs
of vertical angles are formed?
3. Select and measure the vertical angles in the given
figures.
m. Supplementary Angles
1. Any two angles whose sum is 180°, or a straight angle,
are supplementary angles.
Each angle is the supplement of the other.
2. When one straight line mee\,a aivo\)aet ^Vx^y;^ Xjcafc ^^
pair of supplementary angles is ioxraad.
74
MODERN JUNIOR MATHEMATICS
^'fBcT-ri*
3. How many degrees are in two supplementary angles?
How many right angles does their sum make? How many
straight angles?
4. If Z A contains 120°, how many degrees are in its sup-
plement, LBi
6. If two straight lines intersect, four pairs of supple-
mentary angles are formed.
Find the pairs of supplementary angles in
each of these figures.
6. (a) With your protractor draw the follow-
ing angles,
(b) Compute and draw their supple-
ments.
(1) (2) (3)
(4)
120°
45°
30°
90°
150°
7. (a)
(b)
22J'
15°
115°
65°
185°
117°
29°
81°
122°
f rt.
|rt.
Z
Urt.
Z
frt.
z
l*rt.
z
Irt.
z
Cc)
In the figure of the parallels, page 72, what is the
sum of Z 2 + Z 4?
Since Z 2 = Z 6, we may substitute Z 6 for its
equal, Z 2, in the equation, Z2+Z4 = 2rt. /^.
.-. Z 6 + Z 4 = 2 rt. A.
What kind of angles are A 6 and 4?
PARALLELS CUT BY A TRANSVERSAL 75
8. Complete the following statements correctly by filling
the blanks with either the word "equal" or "supplementary."
(a) K two straight Unes intersect, the vertical angles
are .
(b) If two parallel lines are cut by a transversal,
(1) the alternate interior angles are .
(2) the alternate exterior angles are .
(3) the corresponding angles are .
(4) the two interior angles on the same side of
the transversal are .
(5) the exterior angles on the same side of the
transversal are .
(c) In the figure of the parallels, page 72, find all the
other angles when Z 2 = 60°; when Z 5 = 110°;
when Z 3 = 45°.
D. PRACTICAL USE OF PARALLELS
1. To divide a line into equal parts.
76
MODERN JUNIOR MATHEMATICS
(a) Draw a series of seven ||s that are equal distances
apart, or use a sheet of paper ruled at equal in-
tervals.
(b) Across the series draw several transversals not
II to each other.
(c) With your compass measure the different parts
into which each line is divided. How do the
parts of the same line compare?
(d) If one wishes to divide a line 5" or 10" long into
five equal parts, it is easy enough, for it can be
done with a ruler or tape measure. But it is
not so easily done if the Une is 4" and a frac-
tion. By using parallels it may be done accu-
rately.
(1) Number the ||s 0, 1, 2, 3 . . . .
(2) Place one end of the line on and the other
end on the 5th ||, as the line AB.
(3) Mark where each of the other parallels crosses
the Une.
(4) These marks divide it into five equal parts.
2. On page 70 we saw how a carpenter could divide
a 12-inch board into three strips. Suppose he wishes to
divide it into five strips of equal width.
He places his square across the board in such a position
that he can mark ofif five equal parts, as at 4, 8, 12, 16, and
20. (By placing it at another angle he might use 3, 6, 9,
12, and 15.)
PRACTICAL USE OF PARALLELS 77
He moves the square to another position and repeats the
process. Through the corresponding marks he draws
straight lines as guides for sawing.
(a) Show that this process is correct.
(b) Could he use marks 4, 8, 12, 16, and 20 for one set
and 3, 6, 9, 12, and 15 for the other? Explain
clearly. Could he use marks 2, 4, 6, 8, and 10?
3. To divide a line into any number of equal parts with
compass and ruler.
To divide AB into five equal parts.
(a) Draw AC making an Z with AB.
(b) On AC mark ofif five equal parts of any convenient
length. From X, the fifth point, draw BX.
(c) Through each point on AX draw a line || to BX.
(d) These ||s divide AB into 5 equal parts. Explain.
4. (a) On squared paper draw any scalene triangle ABC.
Draw an altitude CD, bisect it at 0, and through
the point of bisection draw a line GF parallel to
the base.
(With protractor draw GF JL to CD at 0.)
Draw GH and FE parallel to CD.
Cut out the A so drawn and by folding carefully
along GFy FE, and GH, show the
□ EFGH = ^ A ABC.
78
MODERN JUNIOR MATHEMATICS
Show that CF
CG
AH
DE
AAHG
BF
AG
HD
EB
AGOC
(b)
(What part of □ GHDO does each A make?)
Show that A COF = A FEB.
(What part of □ DEFO is A EBFf Is A OFCf
What part of A ABC is in the □ HEFG?)
If a line is drawn parallel to one side of a triangle,
bisecting another side, what does it do to the
third side?
5. Draw any scalene triangle. Bisect each side and
join the midpoints.
c Cut out the A and cut
ofif one of the smaller A.
Try to fit it on the other
A. How do these lines
joining the midpoints
divide A ABCf
Compare A 1 with Z
^ D B 2. What is the relative
position of AB and FEf What other lines are parallel?
CHAPTER SEVEN
QUADRILATERALS
A. CONSTRUCTION OF QUADRILATERALS
At first we drew squares and rectangles on squared paper.
Now we shall learn to draw them with compass and ruler.
1. To draw a square,
(a) Suppose AB is the side of the square.
AtAdrawAXl.toAR With
compass measure AD = AB,
With B and D as centers and
an opening = AB, make two
arcs intersecting ate. Draw
BC and CD. Then ABCD
is a square.
(b) Test and see that it has all the
necessary features of a square.
2. To draw a rectangle,
(a) On the sides of a rt. Z lay off AB the length and
A B
AD the width of the □. With D as a center
and an opening of the compass = AB, draw an
^^ above B, With B as a center and an open-
ing = AD, draw an arc intersecting the first
one at C, Draw BC and CD,
(b) Show that ABCD is a rectangle.
79
80
MODERN JUNIOR MATHEMATICS
3. To draw a parallelogram.
D
A B
(a) If AB and AD are the two sides of the O let
them meet at any Z .
(b) How can you find point C and complete the O?
4. To draw a parallelogram equal to a given parallelogram.
D ^C P uo
M
A B
Let ABCD be the given O.
Take MN = AB.
At M make an Z = Z BAD, and take MP = AD.
How can you complete the O?
B. THE SQUARE
I. Relation of Lines and Angles
1. Draw and cut out a square.
2. Fold carefully on two diagonals.
D
C
\
/
X
\
A
B
2 C
^
M
%
V
P X
N
3. What does each diagonal do to the A at each comer
or vertex? How large is each Z?
What is the size of Z E?
THE SQUARE
81
1. How does each diagonal cut the other?
6. Are they perpendicular to each other?
6. How many A are formed?
7. How do they compare with each other?
8. Bisect each side of the square and fold across.
9. How does the A ABE compare with the D PXNOf
10. MN and PQ, the lines joining the midpoints of the
sides of a square, are its medians or midj<nns. Sometimes
these lines are called the diameters.
11. (a) Join the ends of the medians
of a square in succession.
(b) What kind of %ure is formed?
(c) What part of the original square
does it contain?
82
MODERN JUNIOR MATHEMATICS
12. By drawing medians and diagonals of a group of
squares and by shading some of the small triangles or other
parts, very pretty designs may be made. These are used
principally in linolemn and tile floors.
(a) Copy the given designs on squared paper or make
original ones.
C. THE RECTANGLE
I. Relation of Lines and Angles
1. Draw and cut out any rectangle. Fold on both diag-
onals. How does each diagonal divide the □? (See page 36.)
C 2. Cut out A AOB
and BOC and fit them
on A COD and AOD
respectively.
3. Are the diagonals
equal? Are they ± to
each other?
4. Do they bisect
B each other?
5. Do the diagonals bisect the angles at the vertices or
corners? In what kind of rectangle is this done?
6. What kind of A is AOB? A BOC? A COD? AAOD?
7. Find all possible pairs of equal angles in the figure.
8. (a) Connect the midpoints of the opposite sides of a
rectangle.
(b) What is the name of these lines?
(c) How do they divide the rectangle?
THE PARALLELOGRAM
83
9. (a) Join the ends of
the medians
in succession.
(b) What kind of
figure is
formed?
(c) What part of
the original
rectangle does it contain?
D. THE PARALLELOGRAM
I. Area
1. (a) Draw a O ABCD. Draw AE and BF ± to AB,
What is the shape of
ABFEf
(b) Any JL between the
bases of a O is its
aUityde.
(c) Cut off the A BFC
and fit it on A ADS.
(d) How does the size of the O compare with that of
the □? Compare their bases; their heights.
(e) What is the formula for the area of the rectangle?
Of the parallelogram?
(f) If AB = 15 and 1 /
B
zoo^
AE = 8, what
is the area of
□ ABFEf
Of O ABCDf
(g) Draw several Z17
and show that
their areas are
equal to the
U] with same dimensions.
2. (a) Two roads intersecting at an Z of 35° divide a
84 MODERN JUNIOR MATHEMATICS
man's land into two tracts as in the figure.
Find the area of each part A and B.
(b) Will each part require the same amount of
fencing?
3. Two fields in the shape of parallelograms have equal
bases and equal altitudes. One has an angle of 50° between
two sides, and the other has 75°.
(a) Which has the greater area?
(b) Which requires more fencing?
/20rd ^2^^
/.OOrd
/ 7
4. Figures A, B, and C represent three fields, each of
whose perimeters measures 400 rods. It is said that pioneers
traded land with the Indians on a basis of perimeters. An
Indian had plot C and was offered his choice of A and B.
Which offer should he take, if either?
Explain answer.
5. (a) A boy has his choice of three lots for use as a war
garden. Each of these takes 20 rods of fencing.
The first lot is in the shape of a square; the
second is a rectangle; the third, a rhomboid.
Which shall he choose in order to have the largest
garden?
(b) How large is the garden measured in square rods?
in square feet?
Is its width longer or shorter than the width of
an ordinary city lot?
(c) Draw to any convenient scale the plan of the gar-
den, allowing 15 inches all around it inside the
fence, and allotting \ to tomatoes, | to cabbage,
and I to small garden truck, as radishes, lettuce,
etc., and the remainder to potatoes.
THE PARALLELOGRAM 85
(d) How many tomato plants must be set out if the
rows are to be 18 inches apart and the plants
in each row are 15 inches apart?
(e) How many seed potatoes must the gardener buy?
(f) If you have had a garden of your own, you may
calculate a fair yield from such a garden.
(g) How much larger is the chosen garden than the
rectangular one, if its length is four times its
width?
n. Relation of Lines and Angles
1. (a) Draw any parallelogram.
(b) Cut out the figure and
fold on one diagonal.
Do the two triangles
fit?
(c) Cut along the diagonal.
Now can they be made to fit?
(d) How does the diagonal of a O divide the figure?
(e) How do the opposite sides of a O compare in
length? in position?
(f ) How do the A at the opposite corners compare in
size?
(g) If Z A = 60°, find the size of A B, C, and D.
(h) Draw a O and its two
diagonals,
(i) Are the two diagonals equal?
Are they ± to each other?
(j) Do they bisect each other?
(k) Do they bisect the A at the vertices?
(1) Do they make any pairs of equal triangles?
(m) Is any one of these A isosceles?
(n) Find all possible pairs of equal angles.
86 MODERN JUNIOR MATHEMATICS
in. The Rhombus
1. (a) Draw an equilateral parallelogram.
What is the name of this O?
(b) Fold on the diagonal.
Do the two triangles fit?
(c) Draw the other diagonal.
(d) Are the two diagonals equal?
(e) Are they perpendicular to each other?
(f) Do they bisect each other?
(g) Do they bisect the A at the vertices?
(h) Do they make any pairs of equal A? Name them.
(i) Are these A isosceles?
(j) What kind of A is ABC? BCD?
(k) If Z BAD is 60°, find the size of all other A in
the figure (15 more in all).
£. SUMMARY
In which of these quadrilaterals
1. are the diagonals equal?
, 2. do the diagonals bisect each other?
3. are the diagonals perpendicular to each other?
4. do the diagonals bisect the angles at the vertices?
5. do the diagonals form pairs of equal A?
6. are these pairs of A isosceles?
CHAPTER EIGHT
MEASURED AND UNMEASURED LINES
A. PROBLEMS FROM RECTANGLES
I. Parentheses
1. Draw a rectangle 3" x 2". Find its area.
(If more convenient, draw to scale |" to 1".)
2. (a) Add two inches to the length of this same rec-
tangle. What is the
area of the added
part?
S otnn = 1 xw
= (3 + 2) X 2
= 6 + 4
= 10 sq. in.
(b) Such an expression as (3 + 2) x 2 means that
the smn of 3 and 2 is to be multiplied by 2. It
may be written thus: 2 (3 + 2) without the mul-
tiplication sign and is read ''two times the
quantity three plus two."
(c) Its value may be found by adding the 3 and 2 and
then multiplying the sum 5 by 2, which gives
10; or each part of the quantity may be mul-
tiplied and the partial products added.
3 + 2
3 + 2 = 5
X 2
X 2
6 + 4 = 10
10
3. What is the perimeter of the n?
Pa = 2il + w) or Pa = 2l + 2w
= 2 (3 + 2 + 2)
= 2(3 + 2) +2-2
= 6 + 4 + 4
= 10 + 4
= 14 inches
= 14 inches
87
88
MODERN JUNIOR MATHEMATICS
4. (a) Find the area of a rectangle 5" x 3".
(b) Express its dimensions if the length is increased
byl"; by 2"; by 3".
(c) Find the area of each.
(d) Find the perimeter of each.
5. (a) Estimate the dimensions of a page in your book.
Estimate its area.
(b) Then measure carefully and compute the area
from the exact measurements.
(c) Suppose the page were 2" longer; how large
would it be?
6. (a) Draw a rectangle 3" x 2". Increase the length
by 2" and the width by 1".
2 -
1
ft
— 1 " 1 1
' 6s<f.in
1 f
Zsq.in-
•
3 sq.m.
1 1
= (3 + 2) X (2 + 1)
= 5x3
= 15 sq. in.
(b) Into how many small [S is the figure divided?
(c) What is the area of each small □?
(d) What is the sum of these areas?
(e) The expression (3 + 2) x (2 + 1) means that the
sum of 3 and 2 is to be multiplied by the sum
of 2 and 1. It may be writtew mthovit tKe
PROBLEMS FROM RECTANGLES 89
multiplication sign, as (3 + 2)(2 + 1), and it is
read, *'the quantity 3 plus 2 multiplied by the
quantity 2 plus 1."
(f) The multiplication may be made in two ways.
3 + 2
or
3 + 2 = 5
2 + 1
2 + 1=3
6 + 4
15
3 + 2
6 + 7 + 2 =
15
The first form
shows
the areas of the
small
rectangles.
7. What is the perimeter of the rectangle given
in ex-
ample 6?
P □ = 2 (Z + 1/;)
or P
'□ = 2Z + 2w;
= 2 (3 + 2 + 2 + 1)
= 2(3+2) + 2(2 + l)
=6+4+4+2
= 2 (5) + 2 (3)
= 16 inches
= 10 + 6
= 16 inches
8. (a) Find the area of a □ 7" by 4".
(b) Express its dimensions if the length is increased
by 3" and the width by 2".
(c) Find its area.
(d) Find its perimeter.
9. (a) Suppose your book were 2" longer and 1" wider,
then estimate its area and perimeter,
(b) Measure carefully and compute both area and
perimeter.
10. (a) Estimate the dimensions and area of your school
room.
(b) Measure and compute its area and perimeter.
(c) Suppose it were 5 ft. longer and 3 ft. wider;
what would be its aiea awdi ^TvmfcXfc^.
90 MODERN JUNIOR MATHEMATICS
11. Find how much varnish would be required to put two
coats of varnish on the floor and wainscoting. (If your
room has no wainscoting, assume one 2J ft. high.)
12. (a) Measure the space necessary for each school desk
and for aisles.
(b) Would it require more or less space to have the
desks face at right angles to their present posi-
tion, and how much?
(c) Could more or fewer seats be placed leaving the
same width of aisles between the rows and
around the room?
13. (a) Measure five other rectangular objects in the
school.
(b) Estimate their areas and perimeters.
(c) Then compute with their dimensions slightly in-
creased.
1.
n. Ratio
(a) A flower bed is 25' x 10'.
longer is it than wide?
Z 25 5 ^,
(b) This statement means that
zs'
How many times
the length 25 has
the same rela-
tion to the
/d
width 10 as 5
has to 2. In
other words, the
length is 2\
times the width.
I 5
The statement - = o ^aay be read in two ways:
" the ratio of the length to the width is the same
as the ratio of 5 to 2,** or " l\sto'U) aiS»^\^\,ci*2i/'
PROBLEMS FROM RECTANGLES
91
(c) If we asked what is the relation between the
width and the length, we would turn the frac-
tion upside down.
I " 25 " 5
This means that the width is ^ of the length.
(d) The comparison or relation between two num-
bers expressed as a fraction is called a ratio,
(Ratio is the Latin word for relation.)
(e) To find the ratio between two quantities, meas-
ure them in the same way, i.e. by the same unit
of measure, and write these measures as a frac-
tion reduced to lowest terms.
(f ) What is the ratio between the length of your desk
and that of your teacher's?
(g) Measure your desk. Perhaps it is 24" long.
Measure your teacher's desk. It may be 3^ ft.
long. You cannot compare these unless both
are measured by the same unit. Measured in
inches their ratio is ff = ^. Measured in feet,
2 4
their ratio is ^ = =•
2. (a) Two gardens are 30' x 50' and 60' x 120' re-
spectively. How many times longer is the
fence surrounding the second than that of the
first?
(b) If the first fence cost $8.00, what will the second
cost?
6d
B
A
92 MODERN JUNIOR MATHEMATICS
Solution of (a).
P of A = 2 (Z + w)
= 2 (50 + 30)
= 2(80)
= 160
Pb ^360^9
P^ 160 4
P of B = 2 (Z + w)
- 2 (120 + 60)
- 2 (180)
»360
Solviion of (b).
To find the cost of the second fence we must find
the number of dollars that will have the same
ratio to 8 that 9 has to 4.
Let n = the nimiber of dollars in the cost of the
second fence.
Then ^ = ratio of these two costs.
o
But this ratio must equal the ratio of the two
perimeters, which is j. Therefore, g = t
Multiply both sides of this equation by 8.
(2 Q If equals are multipUed by
^ j = B X 7 equals the products must be
' equal,
n = 18
Therefore the cost of the second fence is $18.
Proof: Does ^ = |?
n 9
3. Note: (a) - = - is an equation.
8 4
(b) To solve an equation is to find the value of the letter
which stands for the unknown number.
(c) To solve this equation we chose to multiply both
sides by 8. Why? We wanted to get rid of the
denominators 8 and 4. By multiplying by the
lowest common denominator, we got an equation
without fractions.
PROBLEMS FROM RECTANGLES 93
(d) An equation is like a balance. Therefore we may
add the same number to both sides, subtract the
same number from both sides, or multiply or divide
both sides by the same number without destroying
the balance.
4. A room is 18 feet long. The ratio of the width to the
length is f . Find the width of the room.
SolvUon. Let w = number of feet in the width of the
room.
^ 11 = i
2 9
36(g) = 36x? xby36. Why?
w = IZ\ ^ by 2.
Therefore, the width = 13^ ft.
Proof: Does rrJ- = t?
18 4
5. The dimensions of two rectangles have the same ratio.
The first rectangle A is 9' x 15'. The second, S is 25' long.
How wide is it?
Therefore, in □ S, ^r^ =7^
25 15
3w; = 45
It; == 15 -5- by 3.
Therefore, the width of □ B = 15 ft.
Does if = ^?
94 MODERN JUNIOR MATHEMATICS
6. A room is 15' x 18'. The rug on the floor is 12 ft.
wide. How long must it be to have the same ratio between
its length and width as between the dimensions of the room?
7. On a map a distance of 150 miles is represented by a
line 1^" long. Two cities are 5" apart on the map. What
is the distance between them?
8. In a geography find a map and read the scale. Meas-
ure as accurately as possible the lines between four pairs of
large cities. Calculate their distances.
9. Draw an angle of 45°. What is the ratio between it
and its complement? Between it and its supplement?
10. Z A is 30°. Find the ratio between Z A and its
complement. Between Z A and its supplement. Between
its complement and its supplement.
11. Measure the length
and width of the □ ABCD to
the nearest tenth of an inch.
What is the ratio of the
length to the width?
Find its perimeter.
12. Measure your school
room correct to the nearest
half-foot. Draw a plan of it
on the scale of 5' to 1".
Locate the teacher's desk and the rows of pupils' desks.
13. Draw a diagram of your war garden on a scale of 10'
to 1". (Use 20' to 1", if more convenient.)
14. Draw a plan of the first floor of your house on a
scale of 10' to 1".
15. This diagram of a large club house is drawn on a scale
of 100' to 1". All angles are right angles.
(a) What is the width of the street? Of the pave-
ment?
A B
PEOBLEMS FROM RECTANGLES
Street
m-
(b) What are the dimensions of the plot?
(c) Room I is a reception hall.
Room II is a dining hall.
Room III is the kitchen.
Room IV is the serving room.
What are the dimensions of each?
(d) What is the area of each?
(e) What is the area of the terrace?
(f) What part of the plot is covered by the building
including the terrace?
(g) What is the ratio of the width to the length of
the plot?
16. In some newspaper or magazine, find some floor plans
of houses. Some will give the dimensions. Others will
give the scale of the drawing.
From the dimensions find the scale, and -wa ^"SKa..
96 MODERN JUNIOR MATHEMATICS
m. Graphs of Ratios
1. (a) Ratios are used practically every day by most
people, but in newspapers and magazines pic-
tures are made of them, because they are more
effective than the numbers. These pictures are
called graphs. Graph comes from the Greek
word which means to write or draw.
(b) The following paragraph and graph is taken from
one of the U. S. Food Administration bulletins.
Milk is the chief food for lime. It is much richer in it than other
common foods. These lines stand for lime, the top one for the lime
in a cup of milk, the others for the lime in a serving of some other
foods. Notice how much more there is in milk than in the others.
Amount of Lime in
1 cup of milk
I cup carrots
I I
legg
1 I
2 slices of bread
O
Milk is the cheapest food for lime. Buy milk. You and your
children need its lime.
How much more effective is that long line for
milk and the short lines for egg and bread, than
for one to be told that the amount of lime in a
cup of milk is almost 6| times that in an egg?
(c) Measure each of these lines to the nearest tenth
of a centimeter and find the ratio of the lime
in milk to that in the carrots, egg, and bread.
2. The following graph was taken from a school paper,
(a) If this graph is correctly drawn, the bar repre-
sented by $250 should be i of the $500-bar.
PROBLEMS FROM RECTANGLES
(b) Measure carefully to see if that ratio holds.
(c) Test also for $750 and JIOOO.
(d) Compare the $200 and $600 bars.
Dobs a High School Education Pat?
^p;H SCHOOL EDUCATION PAYS
^K YEARLY INCOME
iga
■ HIGH SCHOOL
■^ TRAINING
AGE
NO H. S.
TRAINING
HhN HIGH SCHOOL
^IN HIGH SCHOOL
$500 ■
750 IH
1.000 ■■
F 1,150 ^mm
1.550 ^H^H
14
1 $200
16
I 250
■ 350
■ 470
^600
18
20
22
24
25
^&88
:- J7,337 «5«=: TOTAL $5,112
1 K SCHOOL TRAINED BOYS-WAeCS S3.50 PER DAY
NO H. SCHOOL TRAINING-WAGES »l.50 PER DAY —
r ^ „„..,,. ®. — ^™».„
k:L,... -. : ^ -^_
How much is your head worth? It is worth as much as you put m
it. How much is your body worth? Ooe dollar and fifty cents a day.
That is, from your head down you are worth lesa than two dollars
98 MODERN JUNIOR MATHEMATICS
The above statistics are taken from data gathered by the Bureau
of Education of the United States Government.
At the age of fourteen, a boy in high school is not usually earning
anything. Often, by working after school, he earns enough to buy his
clothing and books. A boy not in school can earn about two hundred
dollars a year, which must pay all of his expenses.
A boy in school at the age of sixteen is making himself more efficient.
A boy sixteen years old not in school.eams only two himdred fifty dol-
lars a year.
The average boy graduates from high school at the age of eighteen.
Then he is able to earn at least five himdred dollars, while the boy
without a high school education is now earning only about three hun-
dred fifty dollars a year.
By the time a boy with a high school education has reached the age
of twenty, statistics prove that he is able to earn seven hundred fifty
dollars; while the boy with whom we are comparing him earns about
four hundred seventy dollars.
The salaries of each are gradually increased until they are each
twenty-five years old. By that time the high school graduate receives
a salary of about one thousand five hundred fifty dollars a year, while
the man with no high school education earns a salary of about six hun-
dred eighty-eight dollars a year.
Within the thirteen years from the time they were fourteen imtil
they were twenty-five, inclusive, the high school student and graduate
has earned seven thousand three hundred thirty dollars. The man
working for thirteen years and having no high school education has
earned five thousand one hundred twelve dollars.
Does a high school education pay?
3. Percentage as a ratio.
When we say that 31% of a person's diet is grains or
cereals we mean that the ratio of the quantity of cereals
that he eats to the total quantity of his food is equal to
the ratio of 31 to 100.
From the following graph read the per cent that each item
of food is of his diet.
PROBLEMS FROM RECTANGLES
(a) The pictures of the ratios are sometimes shown
in a circle graph instead of in bar graphs,
(b) Around the center of a circle there are four right
angles or' 360°.
(c) Each right angle is i of 360"
and 25 % is i of 100%. There-
fore each right angle repre-
sents 25%.
(d) If the amount of cereal in Ex, 3
is 31%, how large an angle
will represent the cereal?
Let n - the number of degrees in the angle,
^'•'366 = 100
10 „ 31 36
aeetr X 2gg - j^ X 3600 xseoo
10 n = 1116
n = 111.6 + 10
Therefore the Z for cereal = 111.6°
MODERN JUNIOR MATHEMATICS
(e) Draw a lai^ circle. At the center make an angle
as nearly 111° 36' as can be made with your pro-
tractor. Shade this angle in some way and
mark it cereals.
(f) Adjacent to the angle make other angles repre-
senting the correct per cents for the other
articles of food. Shade each angle in a differ-
ent way, either with lines running in different
directions, or with different kinds of lines or
with cross bars.
Sampi^b of Sbadikq
5. In your geography and science books you will find
many graphs. Find several and explain their meaning.
6. (a) The following circle graph shows the approxi-
mate distribution of the people of the United
States according to their fields of work,
(b) By measuring the angles find the per cent of
population in each occupation.
PROBLEMS WITH UNMEASURED LINES 101
(c) The sizes of four anglra with fractional number
of degrees are given. Others can be measured
accurately. The A for transportation and
mining are each 10.8°. The A for clerical and
public service are each 7,2°.
B. PROBLEMS WITH UNMEASURED LINES
I. Perimeters and Areas
1. (a) Draw a line of any unmeasured length. Draw s
square using this line as a side.
102 MODERN JUNIOR MATHEMATICS
(b) Each pupil will probably draw a line different in
length from the others, but each line will have
a certain number of inches or parts of inches
in its length, although we do not know just
what the number is. In place of the number
we do not know, suppose we use the first letter
of the word number and say the side of the
square is n inches long, or more briefly,
AB = n inches.
AD = n inches.
(c) Now if one happens to draw his line 2", then the
area of his square is 2^ or 4 square inches. But
no matter how long the lines are, the area of
each square is nn, or w^, and the perimeter of
each is 4 n. We see then that
(1) n-n = 71^
and (2)n + n + n + n = 4n
What do you call the figure 2 used in n^ in the
first statement? What does it show? What
do you call the figure 4 used in 4 n in the second
statement? What does it show?
(d) We see that n^ stands for area and its picture is a
square; but 4n stands for length and its pic-
ture is a line.
n + n -h n + n
Perimeter =■ 4 n. (Scale \)
(e) Measure your own line and find your own values
of n^ and 4 n.
(f) We might have used any other letter instead of
n, as m, p, x, y, 2, or any other.
2. If the picture of n^ is a square whose side is n, can you
draw a picture of 4/1^? Remember that {2ny = 2n-2n
= 22. n2 = 4n2.
PROBLEMS WITH UNMEASURED LINES 103
3. Draw a picture of § n, 2 n, f n, 1§ n, 9 n^.
4. What line can you draw in the last picture to make
one of 4J w^?
5. Use a different length of line (any that is not too
large) for each different letter, and then draw pictures of 2 m,
p2, m H- p, x2, y + 2, 3 X, 2 X + 3 y, m + 2 n + 3 p, J X + I ?/.
6. (a) Draw w^. Add 2 cm. to the length.
71
-t
n
n^
4-
1
2n
1
(b) What is the shape of the new figure?
(c) What is the area of each part?
(1) We saw in the preceding exercise that 2n
meant n + nand then its picture was a line.
In this figure we see that 2 n may mean
2 X n and then its picture is a rectangle.
(d) Find the area and perimeter of the new figure.
Sa =^ Ixw
= (n + 2) n
= n2 + 2 n
Pm =
2{l + w) or Po
= 2 (n + 2 + n)
= 2 (2 n 4- 2)
= 4n + 4
= 2l + 2w
= 2 (n + 2) + 2 n
= 2n + 4 + 2n
= 4n 4- 4
104
MODERN JUNIOR MATHEMATICS
7.
8.
e) Measure n in centimeters and find the area and
perimeter.
f ) What is the imit of measm^ of n* + 2 nt Of
4n + 4?
a) Draw any line x cm. long.
b) Draw a rectangle x + 3 centimeters by x centi-
meters.
c) Compute its area and perimeter.
d) Measure x to the nearest tenth of a centimeter
and find the ratio of its width to its length.
e) Find the numerical values of the area and per-
imeter.
a) Draw n\ Increase its length 3 centimeters and
its width 2 centimeters.
n
77
z-
2n
3r^
4-
+
(b) What is the area of each of the four small rec-
tangles? What is their sum?
(c) The sum expressed as n2 + 3n + 2n + 6 is said
to have four terms, n^ is one term; 3 n is an-
other; and so are 2 n and 6 attieit tftima-
PROBLEMS WITH UNMEASURED LINES 105
(1) An expression having only one term is called
a monomiaL Literally the word means
"one named/'
(2) An expression having more than one term is.
called a polynomial. Literally it is "many
named."
(3) A polynomial having only two terms is called
a binomial; and one having only three
terms is called a trinomial.
(4) Derivations:
Mono — one
Bi — two
Tri — three
Poly — many
+ nomen = name
Monomial
Binomial
Trinomial
Polynomial
You are familiar with the prefixes mono- and
bi' in the words monoplane and biplane or
bicycLey and with the prefix tri- in the word
tricycle or trisect.
The word nomen or name is found in our word
nominate^ which means to name far an office.
(d) We find that the sum of the four rectangles gives a
polynomial of four terms. Can you combine two
of the terms so as to make a trinomial out of it?
(1) Sa^Z-w;
= (n + 3)(n + 2)
= n^ + 5 n + 6
(2) Process by multiplication.
n + 3
n + 2
n2 + 3 n
+ 2n-h6
n^ + 5 n + &
106
MODERN JUNIOR MATHEMATICS
(3) Find the area or S as the sum of two small
rectangles.
S □ = Ai (n + 3) + 2 (n + 3)
= n2 + 3n + 2n + 6
= n^ + 5 n + 6
(e) Find the perimeter.
Pc3 = 2(l + w)
= 2(n + 3 + n + 2)
= 2 (2 n + 5)
= 4 n + 10
or P ^2l + 2w
= 2 (n + 3) + 2 (n + 2)
= 2n + 6 + 2n + 4
= 4 n + 10
What is the name of the unit of measure of the
perimeter?
(f ) Measure n in centimeters and find the value of the
area and perimeter. Find the value of each
term separately before adding.
(g) (1) Draw another unmeasured line m inches long.
(2) Draw a square on the line.
PROBLEMS WITH UNMEASURED LINES 107
(3) Increase its length by 3 inches and its width
by 1 inch.
(4) Find the area of the square; of the rectangle.
(5) Find the perimeter of the square; of the
rect-angle.
(6) Measure your line m to the nearest tenth of
an inch and find the numerical values of the
areas of the square and of the rectangle.
(7) Find the numerical values of the perimeters
of both figures,
(h) (1) Draw squares on four other unmeasured lines.
(2) Increase the length and width of each by dif-
ferent numbers of inches.
(3) Find the area and perimeter of each figure.
(4) Find the numerical measure of each area and
perimeter.
9. (a) Draw an immeasured line x inches long.
{h) Draw a square 2 x inches on a side. What is its
area
2.x
ZX
- - -1 ■■■— —
2X
(c) Add 1 inch to the length of the square. Find the
area and perimeter of the new figurt\
(d) Measure the line x and find the value of the ex-
pressions for the ai-ea and tlie jxM'inieter. To
find the vahie of 4 j '-, squait^ the measuix^ of x
before multiplying by 4.
^e) Find the ratio of the wkUVv \o W\vi Vwj^Vv.
108
MODERN JUNIOR MATHEMATICS
(f ) Find the ratio of the area of the original square to
the area of the rectangle.
10. (a) Draw a rectangle 2 x + 3 inches long and x + 2
inches wide.
^x 4- 3
X
1
1 r -
3X
. 4.x
6
(b) What is the area of each of the four small rec-
tangles? Express the siun as a trinomial.
1
X
T T"
x(ax+3;
xQix-^s)
(c) Omit one of the dividing lines and indicate the area
as the sum of two rectangles.
(1) Solution of (b).
Sa = IXW
= (2a; + 3)(a; + 2)
= 2x2 + 7a; + 6
Multiphcation
2x +3
x + 2
2 x2 + 3 X
+ 4x + 6
PROBLEMS WITH UXJilEASURED LINES 109
(2) Solution of (c).
S = X (2 X + 3) + 2 (2 X + 3)
= 2x* + 3x + 4x + 6
= 2x- + 7xH-6
11. Find the perimeter of the rectangle given in exam-
ple 10.
12. (a) Measure the line x and find the nmnerical num-
ber of square inches in the area,
(b) Find the number of inches in the perimeter.
(1) Solution of (a).
Z = 2x + 3
w = X + 2
By measurement, x = f in.
X' = (ly - A sq. in.
We found S = 2x« + 7xH-6
- 2 (A) + 7(f) + 6 Subst. i for x
= * + ^ + 6
= 1| + 5i + 6
= 12| sq. in. area of □
Note: Prove this area is correct by multiplying the length in inches
bv the width in inches.
(2) Solution of (b).
In example 11 we found P = 6 x + 10.
By measurement, x = |
P - 6 X + 10
Therefore ( .*. ) p = 6 ( J) + 10
= 4^ + 10
= 14^ inches.
Note: The word therefore is usoii a great many times in mathe-
matical problems. The symbol for it is thnH> small dots put in the form
of ao equilatera] triangle (.'.).
110
MODERN JUNIOR MATHEMATICS
13. Find the areas and perimeters of rectangles whose
dimensions follow.
In each case, measm^ the given line and check the results.
Length
Width
(a)
2n + 3
n + 4
(b)
a + 5
a + 3
(c)
32/ + 4
22/ + 3
(d)
2x + 5
x + l
(e)
2a; + 5
2a; + 3
(f)
a + 4
a + 2
(g)
6 + 7
6 + 1
•(h)
2c + 10
c + 3
(i)
n + 6
n + 6
a)
5a + 3
4a+ 1
(k)
5a + 2
5a + 2
(1)
2n + 5
2n + 5
14. (a) Draw two immeasured lines, x and y centimeters
long, and draw a rectangle of these lines.
X
(b) Find area.
S C2 == Ixw
= X X 2/
= xy
Just as 2 X means 2 times x, so xy means x times y.
The times sign is omitted between two factors
when one or both of them are letters. In
arithmetic 2x3 could not be written 23, for
digits in arithmetical numbers have place value
and the 2 means 2 tens or 2.0.
PROBLEMS WITH UNMEASURED LINES HI
(c) Find perimeter.
P ^2(l-\-w)
= 2{x + y)
or
P ^2l + 2w
=2x+2y
15. (a) Draw a rectangle 3 x centimeters long and 2 y
centimeters wide.
3X
^y
(b) Find the area.
Sen "= Iw
= 3x'2y
= Q xy
How many rectangles the size of xy does it con-
tain?
(c) Find the perimeter.
P =^2{l + w) or
= 2 (3 X + 2 !/)
= 6x + 4 2/
P ^2l + 2w
= 2 (3 x) + 2 (2 !/)
= 6x + 4 1/
(d) Measure x and y in centimeters; find the value
of S and P and the ratio of the dimensions.
16. Find the areas and perimeters of rectangles having
the following dimensions:
Length
Width
Length
Width
(a)
2x
3y 1
(e)
le
\d
(b)
5a
65 1
(0
V
\z
(c)
4 m
3w
(g)
2\a
46
(d)
ix
iy
(h)
1
\
-t^x
112
MODERN JUNIOR MATHEMATICS
17. (a) In finding the product of two factors that have un-
like letters, what do you do with the coeffidentsf
(b) What do you do with the two letter factors?
(c) How do you write the product of two like letter
factors?
18. (a) How does the picture of xy differ from that of x^?
(b) What is the picture of x -h y?
(c) Can the picture of x^ ever be a line?
(d) Can the picture of xy ever be a line?
19. (a) Draw two unmeasured lines, a and 6 centimeters
long, respectively,
(b) Draw a rectangle whose length is 2 a + 3 6 cen-
timeters and whose width is a + 2 6 centimeters.
za '\' 3b
;ib--
a a*
1 1
3ab
4flb
eb*
6 cm. long. (Scale })
(c) What are the dimensions of each of the four small
rectangles?
(d) What is the area of each?
(e) What is the sum of these parts of the large rec-
tangle?
(f) Can you give the sum as a trinomial?
(g) Find the area by multiplication.
Sn = Ziy
2a + 36
= (2a + 3 6)(a + 26)
a+26
= 2 a^ + 7 a6 + 6 62
2 a2 + 3 ab
4- 4 a6 + 6 62
\ 2.0? ^1 ob ^<ci\)^
PROBLEMS WITH UNMEASURED LINES 113
(h) Find the perimeter.
P ^2(l + w) OT P = 2l + 2w
-2(2a + 36 + a + 2 6) =2(2a + 3 6)+2(a + 2 6)
= 4 a + 6 6 -h 2 a + 4 6
= 6 a + 10 6
- 2 (3 a + 5 6)
= 6 a + 10 6
(i) Evaluation.
(1) Area.
By measurement, a = 3 cm.
6 = 2 cm.
•
From these measurements what is the value
of a2? of ab ? of ¥?
So = 2a2 + 7a6 + 662
= 2(32)+7(3)(2)+6(2)
= 2 (9) + 7 (6) + 6 (4)
= 18 + 42 + 24
= 84 sq. cm.
(2) Perimeter.
P = 2i + 2ii;
= 6 a + 10 6
= 6 (3) + 10 (2)
= 18 + 20
= 38 cm.
or
or Z = 2 a + 3 b
= 2 (3) + 3 (2)
= 6 + 6
= 12
ly = a + 2 6
= 3 + 2 (2)
= 3+4
= 7
S ^Iw
= 12-7
= 84 sq. cm.
P ^2(l + w)
= 2 (3 a + 5 6)
= 2 (9 + 10)
= 2 (19)
= 38 cm.
(j) What is the ratio of w to If
20. (a) Find the area of a rectangle whose dimensions are
3 a; + 5 2/ and 2x + 3y,
(b) Find the area in square inches if x « 4 in. and
y = 5 in.
114 MODERN JUNIOR MATHEMATICS
(c) A convenient form is to place the multiplication
and evaluation side by side as follows:
l^ix+ by =3(4) +5(5) = 12 + 25= 37
w = 2x + 3y =2(4) +3(5)= 8+15= _2i
6x^+10xy 111
+ 9xy+15y^ 74
>S = 6x2 + 19x1/ + 151/2 = 6 (16) + 19 (20) + 15 (25)
= 96 + 380 + 375
S = 851 = 851
sq. in.
(d) If the two evaluations give the same number, it is
proven that the multiplication is correct. Such
evaluation is called checking, for by it any error
may be checked.
(e) Correctness of the coefficients alone may be
checked by letting x and y each equal 1. Thus:
I = 3x +51/ =3 + 5 =8
w; = 2x +31/ =2 + 3 =5
6 a;2 + 10 ^2/
+ 9 X2/ + 15 i/2
S = 6 ^2 + 19 xi/ + 15 2/2 = 6 + 19 + 15
= 40 =40
This method of checking does not show any errors
in the letters or exponents.
21. (a) Find the areas and perimeters of rectangles hav-
ing the following dimensions.
• (b) Check the coefficient of Examples (1) to (8) in-
clusive by letting x = 2/ = 1.
(c) Check completely Examples (9) to (16) inclusive
by letting the first letter have a value of 2
and the second a value of 3. (Other values may
be used if desired.)
PROBLEMS WITH UNMEASURED LINES 115
Length
Width
(1)
a-hb
a + &
(2)
2x-hy
x-\-y
(3)
w + 2n
m -hn
(4)
2x + 5z
x-hSz
(5)
a^-2x
a-\'2x
(6)
ia + i6
ia + i6
(7)
ix-\-\y
ix + iy
(8)
3c + 5d
3c+5d
(9)
2a + 6
a+26
(10)
Zx-\-2y
3 a; +-2 2/
(11)
a + 36
a + 36
(12)
a; + 5
x +4
(13)
2x + Z
2a; + 3
(14)
3a; + 52/
3a; + 72/
(15)
3w + 2n
2 m +n
(16)
2o + i6
a + i6
5
(d) Which of these rectangles are squares?
22. (a) Examine carefully the products obtained in the
previous exercise; for instance, Example 15.
/ 3 m + 2 n\
(1) ^(2)<^ (3)
\2m -f- n i/
6 m^ -h 4 mn
+ 3 mn + 2 n^
6 m^ -h 7 mn + 2 n^
(1) (2) (3)
(b) How is the first term of the product 6 m^ obtained
from the first terms of the binomials?
(c) How is the third term, 2 n^, obtained from the
second terms of the two binomials?
(d) How is the middle term fowxvd'?
116
MODERN JUNIOR MATHEMATICS
Such a term is the sum of the two products ob-
tained by multiplying crosswise.
(e) A Uttle practice will enable one to write out the
product of two binomials without showing all
the work of multiplication. Thus:
(1) (3)
(1) (3m + 2n)(2m-fn)
(2) (a + 2 &) (a + 3 &)
(1 + 4) (1 + 6)
(5) (7)
35
' 6 m« + 7 mn + 2 n«
(1) <2) (3)
a2 + 5 oft + 662 Check a=l
1 + 10 + 24 6=2
35
35
23. (a) Find the areas and perimeters of the following
rectangles and check results.
(b) Which ones are squares?
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
a + 1
x + 4
2a; + 3
3n + 4
2a+36
3a + 26
28-\-t
4a + 36
4x + 6
Qx-hy
6x + 5
x + 9
2y-hSz
3a + 10
P + 2g
a + 1
a;+3
x + 1
2a; + l
3n + 4
2a + 6
a + 26
2s-\-t
4a + 56
4x + 6
x-hQy
x + e
x + Q
22/ + 3Z
3a + 10
p + q
Perimeter
PROBLEMS WITH UNMEASURED LINES 117
24. (a) Draw three lines of different length.
(b) Draw a line equal to the
a sum of the three; as,
a + b + c,
b (c) On this Une construct a
square.
c (d) Draw lines at the points
of division, as in the
figure,
(e) Filid the area of each part of the square.
a ■¥ b A
y c .
a^
ah
ao
ah
h^
bo
ac
be
c^
+
c
(1) How many small parts make up the large
square?
(2) How many of these parts are squares?
(3) What special position do these small squares
have?
(4) How many of the parts are rectangles?
(5) How many have a and b as dimensions?
(6) How many have a and c as dimensions?
(7) How many have b and c as dimensions?
(8) What is the area of (a + & + cY ?
(9) Write the area in six terms.
(10) Check with a = l,b = 7., ^tA c -%.
118 MODERN JUNIOR MATHEMATICS
25. What is the area of a square on the sum of three lines
X, y, and z? Draw a figure and check.
26. (a) Three numbers a, 6, and c, can be combined in
three different products, afe, ac, and fee, by com-
bining each one by every other one that follows.
Besides, each can be combined with itself.
(b) What kind of figure does each of the latter com-
binations make? What kind does each of the
former make?
(c) In how many different products can you combine
four numbers, a, 6, c, and df
(d) Represent these by different lengths of Unes,
construct a square on the sum, and divide it
into parts.
(e) How many of these parts are squares?
(f) How many are rectangles?
(g) How many different sized rectangles are there?
How many of each kind?
(h) We find that
(a -h 6 -h c)- = a2 -h 62 -h c2 + 2 a6 + 2 oc + 2 fee and
(a + fe -h c -h (i)2 = a2 + fe2 + c2 -h d2 _^ 2 ofe +
2 ac -h 2 a(i -h 2 fee -h 2 fed -h 2 cd.
Thus we see that a square constructed on the sum of
any number of lines contains a square on each
line plus two rectangles of each possible com-
bination,
27. (a) What is the area of a square constructed on the
following lines?
(b) Check results.
(1) a + 2 fe (6) a + fe + ic + 2/
(2) a + 2 fe + c ' (7) a + fe + 2
(3) a + 2 fe -h 3 c (8) x + 2/ + 2 + 3
(4) 2 a + 3 fe + 4 c (9) 2 m + 3 n + 4
(5) 3x-h2y + z {\^) a\h\z\^
PROBLEMS WITH UNMEASURED LINES 119
n. Finding One Dimension
1. (a) Draw a 6-inch square. What is its area?
(b) Draw a rectangle 4" wide whose area is equal to
that of the square.
(c) How do you know how long to make your rec-
tangle?
(d) Draw another rectangle equal to the square in
size with a width of 3".
(e) If the area of a rectangle and one dimension are
known, how can you find the other dimension?
(f) Write your statement as a formula:
w
to be read, I equals S divided by w,
(g) The line between the S and w is the line of the
division sign (-^) in which the letters have taken
the places of the two dots.
(h) 8 divided by 4 may be written in three ways :
(1) 8 -^ 4, with the division sign.
(2) I, as a fraction.
(3) 2, as a quotient.
(i) The division sign is not often used with letters
representing numbers, but the fraction form is
used instead; as t, which means a -^ b.
Note : It is interesting to know, however, that the fraction
form of showing division is about 500 years older than
the division sign. The different ways of expressing
division as used by various people during the last 1500
years are:
(1) From about 500 to 1200 a.d., the Hindus wrote the divisor
over the dividend with no line between; as , .
b
120 MODERN JUNIOR MATHEMATICS
This form was found in a book written by a Hindu
about 1150 and was probably used several hundred
years before.
(2) About 1000, the Arabs used a straight line in one of
a
three ways; a - 6, a/6, or -.
(3) In 1631, in England, Oughtred used a dot; as, a- 6; and
in 1657 a colon; as a :&.
(4) In 1668, in England, Pell used the division sign, as we
know it and as it is used today in English speak-
ing countries.
(j) Any fraction is an indicated division; as, f , f ,
a a + b x^ xy
(k) Sometimes the division may be performed and one
number be obtained as a quotient; as,
(1) 4 = 2 or - =. ^-^ = 2
(2)^^a: or ^^ *^ = a:
X X f
(3)f = , or f = ^^ = ,
(1) With other numbers the division cannot be per-
formed and we use the expression in the form
--,. 2 15 aa + fcx x
of a fraction; as, 5,
3' 16' V b ' y' x + y'
2. (a) Write the formula for the width of a rectangle,
(b) Translate the formula into English.
3. (a) In each of the following rectangles, the area and
one dimension are given.
PROBLEMS WITH UNMEASURED LINES 12]
(b) Find the other dimension.
SC2
I
w
(1)
144 sq. in.
9 in.
(2)
289 sq. ft.
17 ft.
(3)
30i sq. yd.
5iyd.
(4)
288 sq. ft.
12 ft.
(5)
15 5 sq. in.
3iin.
(6)
12i sq. ft.
Uft.
(7)
289 sq. rd.
12i rd.
(8)
256 sq. in.
32 in.
(9)
4.41 sq. in.
6.3 in.
(10)
72.8 sq. ft.
3.8 ft.
4. If a rectangle contains 12 x^ sq. ft. and is 4 a; feet long,
how wide is it?
(a) SoliUion.
S
~ 4 X
^ i'i'3'f'X
= 3x
\ n is 3 a; ft. wide.
(b) Explanation:
(1) The fraction
12^
4:X
may be reduced to lowest
terms by dividing both the numerator and
denominator by all of the common factors,
just as ^ is reduced to f .
18 ^ 2!-3'g ^3
24 2-2. 2-3 4
122 MODERN JUNIOR MATHEMATICS
(2) Or, the indicated division may be solved as a
3
short division problem as, 6jl8.
3a;
Then ?i^is4£jl2x2.
4a;
(3) To find the quotient 3 x, the coefficient 12 is
divided by the 4 as in arithmetic.
(4) But dividing the x^ by the x is quite different.
We found that
92
3-3=32 ... 2.^3
.ndx.x^x^ ... ^- = .
X
(5) We say that x^ is the second power of x and
x^ is the third power of x, and write the
little 2 and 3 as exponents to show how
many factors have been used.
But we do not use the exponent 1, as x^ to
show that X is the first power, although it is
always understood. In finding the product
of a; X a; or a;^ X a;^ we add the two l^s to get
the exponent of x^.
Likewise x-x-x = a;^'a;^-a;^ = a;^ + i + i = a^
and x^'X = x^-x^ = a;^^^ = x^,
(6) Since we add the exponents of like letter fac-
tors to find the product, we must do the
opposite to find the quotient; that is, we
must subtract the exponent of the divisor
from the exponent of the dividend to find
that of the quotient.
Therefore, x^ -^ x ^^ [x^ -i- x^ = a;^" ^ = a;^] = x
and a;^ 4- a; = [a;^ -^ a;^ = a;^" 1] = a;2
and a;^ -^ a;2 = [a;^~^ = a;^] = x.
Note: The parts in brackets are ©ven here ouly for
the sake of explanation.
PROBLEMS WITH UNMEASURED LINES 123
6. (a) Find the missing measurement of the following
rectangles from the two that are given,
(b) Use the formulas.
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(18)
(19)
(20)
Area
24x2
24x2
108 a2
1171/2
7St
12 xy
25 ab
69 mn
72 ax
72 ax
72 ax
72 ax
Length
8x
12 o
12 6
5a
36 a
No
a {a-\-h)
6 (a + x)
21 a (a + c)
18 a
a + 1
• • • • •
x-\-y
a +x
3 (m + w)
3 (a + c)
Width
4x
3y
5i6
6^
2y
3w
2a
8x
6 a
4x
a
a
3
4m
6. Draw a rectangle whose length is 2 a + 3 h and whose
width is a + 2 6. Find its area.
.-^ 3b
a
-M-
Zb
2^
+
S-Qza-hSbXa-h^b)
^Za^-t7abi-6b^
124
MODERN JUNIOR MATHEMATICS
Given a □, with I = 2 a + 3 &
and w; = a + 2 6
Discussion.
(a) We may write the area in two ways:
(1) S = (2 a + 3 6)(a + 2 6), expressed in factors.
(2) S = 2 a^ + 7 oft + 6 &2, expressed as a product.
(b) If the area is given in factors and one dimension
is known, then the other dimension is easily
fomid.
l^ S ^ (2a-h3b)(a + 2b)
w a 4- 2 6
= 2a + 36
(c) Remember that in this area each factor has two
parts or terms and that the factor as a whole
must be used and not the parts taken separately.
(d) If the area is expressed as a product, then the
second dimension must be found by long
division.
(e) Short division is used if the divisor is small enough,
whether it contains numbers or letters; as,
6 6 a 6 a + 2
18
3 a 18 a2
3 a 18a2 + 6a
(f) Long division is used if the divisor contains more
than one term.
(g) Compare with long division in arithmetic.
31
r 30 + 1 =
= 31
31 961
30 + 1 900 + 60+1
93
31
31
900 + 30
30 + 1
30 + 1
The second illustration is just Uke the first except
that the numbers are expressed with units, tens,
PROBLEMS WITH UNMEASURED LINES 125
and hundreds separated. The second form is
always used in long division of numbers ex-
pressed with letters.
If t represents tens, then t^ will represent himdreds
for
100 = KF = f; and 31 = 30 + 1 = 3 « + 1 and
961 = 900 + 60 + 1 = 9 ^2 + 6 < + 1
3^ + 1 = 30 + 1=31
3< + l 9f + 6t + l
91^ + St
3~< + l
3JJ-_1
Solviion,
Find the length of the rectangle by long division.
Given S = 2a2 + 7a& + 6 62
and w = a + 2b
^ _S ^ 2 g^ + 7 gfe + 6 y
" w a + 26
2a + 3fe .-. Z = 2a + 36
aj-_26j2a2 + 7a6 + 6 62
2a2 + 4a6
3a6 + 6&2
3 gfc + 6 b^
Explanation,
(a) Since a is the first term of the divisor, the term of
the dividend having the highest power of a
must come first; the next lower one, second;
and the term without a must come last.
(b) By dividing a into 2 g^, we get 2 g, which is the
first term of the quotient.
(c) By multiplying both terms of the divisor by 2 g,
we get 2 g2 + 4 g6, which is to be subtracted
from the dividend.
126 MODERN JUNIOR MATHEMATICS
(d) The second term of the quotient 3 6 is found by-
dividing a into 3 ab.
(e) Multiply a + 2 6 by 3 6 and subtract the
product from 3 a6 + 6 6^. There is no remainder.
Problems:
(a) Given S = 2a2 + 7a6 + 6&2
I =2a+3fe
To find w,
S 2a^ + 7ab + &b'
w — — —
I 2a+36
a + 2& .'. w =^ a + 2b
2a_±3bj2a^ + 7ab+&¥
2 a2 + 3 a&
Tab + Qb^
4 a6 + 6 62
(b) Given S ^ Qx^ + I9xy + I5y^
w == 2x + Sy
To find I
, S ^ 6 X" + 19 xy + 15 y^
~ w 2x4-31/
Sx -{- 5y /. Z = 3 a; + 5 y
2x + 3y\ Qx^ + l9xy + 15y^
6 oj^ + 9 xy
10 xy + 15 2/2
10 xy + 15 2/2
(c) Proof: (1) Proof by multiplication.
S = l'W
6 x2 + 19 a;2/ + 15 2/2 = (3 x + 5 y){2 x + 3y)
= 6 ^2 + 19 a:2/ + 15 y^
Product \iy \Tk&^c^\aw.
PROBLEMS WITH UNMEASURED LINES 127
3a: +5y
2x +3y
6 a:2 + 10 xy
+ 9xy + 15y^
6 x2 + 19 xi/ + 15 y^
Product by long multiplication.
(2) Proof by checking.
Let a = 2 and 6 = 3.
6 a;2 + 19 XT/ + 15 2/2
2x + Sy
6 (2)2 +19 (2) (3) +15(3)2
2 (2) + 3 (3)
24 + 114 + 135
4 + 9
273
=3x+5y
= 3 (2) + 5 (3)
= 6 + 15
= 21
13
21 =21
The coefficients alone may be checked by
substituting x = y = 1.
Gx^ + 19xy + 15 2/2
2x + Sy
6 + 19 + 15
2 + 3
= 3a: + 52/
= 3 + 5
5 -^
8 = 8
S
(3) Proof by division. Use w = j,
2 x + Sy
Sx + 5y | 6a;2 + 19xy + 15y^
6x2 + 10x2/
9 X2/ + 15 y^
9 x-y + 15 ^^2
!8 MODERN JUNIOR MATHEMATICS
7. (a) Find the missing dimension of each of the follow-
ing rectangles, (b) Prave your results correct.
20a' + 58n6+42h*
31^ + Ucti + StP
f> n' + 23 nfc + 15 1'
3(* + lliu + 10»'
27 a" + 15 ly + 2 y'
15 a> + 37 oft + 20 6"
32 m' + 16 mrt + 22 n'
9o'+21ob + 16 6'
15 a^ + 26 ly + 8 J/'
a" + 10 o + 21
■ i" + 1 J + ItJ
81 a> + 9 a + i
8i« + 76j:t + 140jt'
5a + 16 6
Sx + iy
5n + 7b
4r + 3B
2.e + 7
9i + 2j/
5e + 4/
8 m + 2 «
30+46
ia+ib
n + 7
i:r + 4
3u+56 + (j
6a + 5i
7 r + 30 t
.2 1 + .3 J/
3ir + 3f
('c^ IWuc/i of the above rectan^ea ate 9«\\iai'c«a''.
PROBLEMS WITH UNMEASURED LINES 129
m. Problems of Finding Dimensions from Areas
1. If you know only the area of a rectangle, can you find
what the dimensions are?
2. (a) If a rectangular flower bed contains 36 sq. ft.,
you cannot tell whether it is
(1) 1' X 36'
(2) 2' X 18'
(3) 3' X 12'
(4) 4' X 9'
(5) 6' X 6'
(6) or other fractional dimensions, as 4§' x 8'.
(b) But if the area is expressed with letters, you can
find the two dimensions by finding its two
factors. Usually only one pair of factors can
be foimd.
3. (a) If a^ + 3ab is the area of a rectangle, we can
easily see that a must be one of the dimensions,
for a is foimd in each term,
(b) By dividing the area, a^ + 3 ab, by one dimension,
a, we find the other to be a + 3 6.
Z = -
S or
w a + 36
a^ + Sab
aja^ + 3ab
a
= a + 36
(c) Instead of writing the problem in the form of a
fraction or division, it is customary to write it
in the form of two factors, or like the formula.
S ^w4
a^ + Sab = a (a + 3 &)
4. (a) If the area is 2 a^ + 6 ob, we see that each term
contains 2 as well as a, so we take out 2 a,
2a2 + 6a6 = 2a(a^^b^
130
MODERN JUNIOR MATHEMATICS
(b) What is the length of such a rectangle?
What is the width?
(c) Find its length, width, and area
when a = 2 ft.
and 6 = 3 ft.
(d) Find these values when a is 10" and b is 2".
(e) What is the ratio of the width to the length in
each rectangle?
6. (a) Find the dimensions of the following rectangles
from their areas.
(b) Check results.
No.
Area
No.
(19)
Area
(1)
xy =
ax -\-bx + CX =
(2)
2x^ + 3xy=
(20)
^by+Qcy -{-10 dy ^
(3)
2x^ + 4:xy =
(21)
3o6 + 6ax+6a2/ =
(4)
5 x2 + 10 X =
(22)
iax + i6x4-icx =
(5)
6 a2 + 9 =
(23)
.2 X* + .4 ax =
(6)
2 w2 -h 6 mn -
(24)
i6^+T^.^=-
(7)
10 c2 -h 15 cd =
(25)
(a+3)»+4(a + 3) =
(8)
21 m^ + 14 mn «
(26)
(m + n)* + 5 (m + n) =
(9)
3 x2 + xz =
(27)
a {m -\- n) -h b (rn -{- n) =
(10)
.5 x2 -1- .25 X2/ =
(28)
J a* + i a5 =
(11)
i a2 -H 1 a =
(29)
ia6 + J 6» + } 6c =
(12)
hx^ + ixy =-
(30)
(c+2)(c+2) + (a + 3)(c + 2) =
(13)
ab -\- ac +(id =
(31)
(w + n) (m + n) + 3 (m + n) —
(14)
xy + y^ + yz^
(32)
J /iB + i W> =
(15)
ib' + ib =
(33)
(a + 5)(a + l) + (a + 5)(a + 3) =
(16)
3ax + 6x» +9 6x =
(34)
x^yz + xy^ + xyz' =
(17)
3(a+6)2 + 6(a + 6) =
(35)
2 a^bc + 4 ofe^c + 6 abc^ =
(18)
(x + t/)2 + 2 (x + t/) =
(36)
.5 a26» + a^c^ + 1.5 a^cP =
6. (a) We have learned how to find the dimensions of a
rectangle from its area when each of its terms
has a common factor.
PROBLEMS WITH UNMEASURED LINES 131
(b) But there is no such common factor in
6 x^ + 19 X2/ + 15 xp-
although that is the area of a rectangle.
(c) On page 115, we learned how to write the product
of two binomials. Reread that page.
(d) To find the dimensions from the area we must
reverse that process.
6x2 + 19x2/ + 15i/2 = ( ? )( ? )
^ ^<tA^ ^^ '^^^ ^ ^ °^^y come from
^^^T^y 6xxx or 3xx2x and
(6x2) +5x1/ 15 2/' may come from
4- 18 xy + (15 y') oy xSy or loy x y.
(6 x2) + 23 xy + (15 y^) ^^^ ^^ *^^^^ ^^ ^^^ possible com-
binations, until the set is
^^^>^^ found that gives the right
2 X + 3 1/ middle term of 19 xy when
10 xy multipUed crosswise and
9 xy added.
19 xy
(3) The first set tried gave 23x1/ as the middle
term and is therefore incorrect.
(4) The second gave 19 xy; therefore 3 x + 5 y
and 2 X + 3 y are the length and width of
the rectangle, respectively.
6 x2 4- 19 xy + 15 y2 = (3 x + 5 y)(2 x + 3 y)
S =^ I Xw
(5) Several other combinations may be tried be-
fore the correct one is found.
(6) Try these other combinations to see if any
gives the coTT^e\, cto^^ ^\Qk^\iL^\».
132 MODERN JUNIOR MATHEMATICS
A
B
C
x + 15y
2x + 15y
Zx'+y
3x + 3^
2x + 5y
D
E
F
x + 5y
6 X + 15 J/
x + y
3 X + 15 J/
2x-+y
(7) If one had to try so many different combina-
tions before finding the right set, it would
be a tedious process, but just a glance at
any of these last six shows that they are
all impossible.
In set A, & X X 15 y = 90 xyy which is entirely
too large. Likewise, in set B, 3 x x 15 i/
= 45 xy.
In the last four, there is still another way to
know that these cross products need not be
tried. In each group one of the dimensions
can be separated again into two factors by
dividing by 3; as, 3 x + 3 y = 3 (x + !/) and
6x + 3i/ = 3(2x + y).
If 3 can be taken out of one of the two fac-
tors of the trinomial, it can be taken out
of the trinomial itself and should be done
first. Thus:
6 x2 -f 21 xy + 15 t/2 = 3 (2 x2 + 7 xy + 5 y2)
= 3(2x-f 5y)(x4-y)
Since it is impossible to take out 3 or any
other common factor from the given area
of 6 x^ + 19 xy + 15 y^, it will be impossible
to take such a factor out of either dimen-
sion. Therefore none oi tVve b^t iowY need
PROBLEMS WITH UNMEASURED LINES 133
be tried. With a little practice one can
easily find the two factors by inspection,
that is, by looking at the problem.
7. (a) Find the dimensions of the following rectangles
from the given areas. Check each.
Given areas are:
(i;
( a* + 8 a + 15
(2)
1 m' + 12 mn + 36 n»
(3)
1 a5* + 10x + 24
(4]
1 3o» + 8a + 4
(5)
46^+86+3
(6;
1 8j/« + 22y + 12
(7)
1 cP+2d + l
(8)
1 c» + 2cd + <P
(9)
1 x' + llax + ZOa*
(lo;
( 49 a? + 14 6a; + 6»
(11)
1 6 a:* + 17 a; + 12
(12;
1 c» + 16 c + 63
(13;
1 49 a« + 7 (rf> + 25 6»
(14:
1 4m» + 16»n-7
(15:
) 4 o« + 8 o + 3
(16:
) 9ai» + 9ij/+2j/»
(17
(18
(19
(20
(21
(22
(23
(24
(25
(26
(27
(28
(29
(30
(31
(32
4 a* + 4 a + 1
4 X* + 4 xy + 2/*
2 a* + 3 ay + 2/2
2x^-\-5xy + 2y*
m* -h 4 mn + 4 n*
2 a* + 5 a + 3
362 + 13 6a; + 12x2
4 c2 + 16 c + 15
4 a2 + 23 a + 15
2/2 + 10 2/ + 25
m« + 16 w + 48
^2 + 26 d + 169
52+2(5) + l
202 + 6 (20) + 3
102 + 15 (10) + 56
202 + 10 (20) + 21
(b) Which of these rectangles are squares?
CHAPTER NINE
REVIEW PROBLEMS
A. PROBLEMS ON PARALLELOGRAMS
1. (a) Measure the A of
the given O,
the sides and
height or alti-
tude,
(b) Compare Z A
with ZC; Z B
with Z D.
(c) What is the sum of Z A + Z Bf
What kind of A are they?
(d) How many pairs of supplementary A can you
find?
(e) What is the sum of ZA+ZB-\-ZC+ZDf
How does this siun compare with the sum of the
A oi sl rectangle?
Compare it with the sum of the ^ of a A.
(f) Draw several different sized Z17 and find the sum
of their angles.
(g) Draw several irregular quadrilaterals and find the
sum of their angles,
(h) To find all the ^ of a O, how many do you have
to measure?
(i) What is the ratio of AD to ABf
(j) What is the ratio of the height to the base?
(k) What is the area and perimeter of ABCDf
134
PROBLEMS ON PARALLELOGRAMS 135
(1) Dtslw the Ji AY smdBX. What kind of quadri-
lateral is ABXY? What is its area? Its
perimeter?
2. Draw a parallelogram with two adjacent sides 2|"
and 1^" respectively. The Z between these sides is 55°.
(a) What is ratio of the two sides?
(b) Draw the height and measm-e it to the nearest
tenth of an inch.
(c) Find the area.
3. (a) Two city streets cross each other at an angle of
110°. The block made by them and two other
parallel streets measm-es 240 feet on one street
and 320 feet on the other,
(b) Draw a diagram of the block on a scale of 100'
to 1".
4. (a) Two sides of a parallelogram are x and y. Their
ratio is f . The included Z is 45°.
(b) Draw the O. Find the perimeter in terms of
the line y.
(c) Draw the height. How does it compare with
the base y?
(d) If the ratio of the height to the base is .53, what
is the area?
(e) Find the perimeter and area ii y = 12 cm.
6. (a) The parallelogram is drawn to a scale of |" to a
foot.
(b) Find its dimensions.
(c) Measure its angles.
(d) Find the ratio of its
sides. Of Z A to
Z B,
(e) Draw its altitude and measure to the nearest
tenth of an inch.
(f) Compute its penmetet ^ivdL\\s> ^\<Ka..
136 MODERN JUNIOR MATHEMATICS
6. (a) A cement pavement 3J ft. wide surroimds a rec-
tangular city block 200' X 300', measured in-
side the pavement,
(b) How many square yards are in the pavement?
7. (a) The city block is in the shape of a parallelogram
300 and 400 feet on the sides, which make an
angle of 55°.
(b) Draw to a scale and find its area.
(c) Find the area of a 4 ft. pavement surroimding
this block.
B. PROBLEMS ON TRIANGLES
1. Draw triangles from the following data.
Measure the sides and angles not given.
(a) AB = 2 J" AC = If" Z A = 55°
(b) AB = 3f " Z A = 35° Z B = 65°
(c) AB = U" AC = 2i" BC - 2f"
2. (a) Draw a rt. A whose perpendicular sides are 1^
and 3 inches respectively.
(b) Compute the length of the hypotenuse.
(c) Measure it to verify your computation.
(d) How many degrees are there in each of the
acute A?
3. (a) Draw a rt. A whose base is 12 centimeters and
hypotenuse is 15 centimeters.
(b) Measure the acute angles and the third side.
(c) What is the Pythagorean Theorem?
(d) This formula may be used to find either leg of a
right triangle as well as the hypotenuse.
We know that 3+4 = 7
.-.3 = 7-4
and 4 = 7 - 3
If a number is subtracted from both sides of an
equation the remainders ate ^tvH e!«\vial. In the
PROBLEMS ON TRIANGLES
137
same way we may apply this axiom to the
Pythagorean Theorem.
a^ + 6^ = c^
aji = c^ — 62
and 62 = c2 - a^
Equals subtracted from
equals give equals.
a = V(^ - 6^ The square roots of the
and b = Vc^ - a^ two sides of an equation
are equal
Solution.
t>= iZ orrv.
Given c = 15 cm.
6 = 12 cm.
To find a.
a^ ^ c^ - 6^
= (15)2 _ (12)2
= 225 - 144
= 81_
.-. a = Vsi
= 9 CTa.
138
MODERN JUNIOR MATHEMATICS
4. (a) Draw a rt. A whose base is 12 inches and hypot-
enuse is 13 inches.
(b) Compute the length of the third side.
(c) Measure it to verify your computation.
6. A telegraph pole 50 feet high is to be steadied by a
wu-e fastened to the pole 30 feet above the ground and to a
stake in the groimd 40 feet from the base of the pole. How
long must the wire be if 1 foot is allowed for fastening it
to the stake and 3 feet for fastening it to the pole?
6. (a) Draw a rt. A whose base is 3" and hypotenuse is 3f ".
(b) Measure the acute A,
(c) Compute the third side.
Solution.
Given c = 3f "
h =3"
To find a.
a
2 _
= c2 - 6^
a =
= (¥-)'
-32
=w- -
9
= W-
144
1«
= «
= v^n
\
= *
Why?
= 2i"
7. (a) Draw the diagonal in a rectangle 5 centimeters
long and 3 centimeters w*de.
3crn
Scm
(h) Measure the diagonal to tYve neaTe^X) YoS^TWiXfcT,
PROBLEMS ON TRIANGLES
139
(c) Compute the length of the diagonal by the Pythag-
orean Theorem and compare results.
8. (a) Draw a square with a diagonal,
(b) Then cP = a^ + a" Why?
cP = 2a^
VdP = V2~a^ Why?
d = aV2
Since d represents the diagonal
and a represents the side
of the square, we find that the diagonal of a
square is equal to the side multiplied by
V2 or by 1.414 +. In other words, the diagonal
is about If times the side of a square.
9. If the side of a square is 12 inches, find the diagonal.
Solution No, 1.
d^ == a^ + a^
= 122 ^ 122
= 144 -f 144
= 288
% d = V288
= 16.97
Pythag. Th.
Sq. Root Process
2' SSW 00 1 16.97
26
329
3387
188
156
a = 12"
Solution No. 2.
3200
2961
23900
23709
d n = aV2 Formula from Ex. 8
= 12 X 1.4142
= 16.9704
140 MODERN JUNIOR MATHEMATICS
10. A baseball diamond is a square 90 ft. on a side. What
is the distance from first to third base?
11. (a) In machine shops, "stock" comes in rods of dif-
ferent sizes with circular ends.
All square rods must be cut
from such roimd stock.
(b) What must be the diameter of
the roimd stock used to cut a
square rod IJ" on a side? A
2" rod? A 2i" rod?
12. A window is 36 feet above
groimd. How far out from the foot
of the wall must a 45 ft. ladder be placed to just reach the
window?
13. The side of one square is 32 in. and that of another is
17 in. What is the side of a square equal to the sum of
these squares?
14. If the sum of two squares is 26 square inches and one
of the squares is 14 square inches, what is the side of the
other square?
16. A shelf 1 foot wide is 5^ ft. from the floor. The
foot of a ladder is placed 5J ft. from the wall. How long
must the ladder be to reach the shelf?
16. A ladder 42 ft. long can be so placed that it will reach
a window 31 ft. above the ground on one side of the street,
and by tipping it back without moving its foot, it will reach
a window 19 ft. above the ground on the other side. Find
the width of the street.
17. In a right triangle, a = 13.6", b = 16.9". Find c.
18. An equilateral triangle is 20 inches on a side. Find
its altitude.
19. The base of an isosceles triangle is 136 ft. Its alti-
tude is 60 ft. How long is its side? What is its perimeter?
20. (a) Draw an equilateral triangle ABC and bisect
Z Cby the line CD.
PROBLEMS ON TRIANGLES
141
(b) How does CD cut AB?
(c) What kind of an angle is Z ADCf
(d) Cut along the line CD and describe the two parts.
(e) Measure the ^i of A ADC.
(f) How does the shortest side compare with the
hypotenuse?
(g) Does this relation hold true between the shortest
side and the hypotenuse of every 30°-60°
right A?
21. How do the two perpendicular sides of a 45^-45° right
triangle compare?
22. If the vertex angle of an isosceles triangle is 80°,
what is the size of each angle at the base?
23. Construct an isosceles triangle with a base 2|", whose
vertex angle is 70°.
24. (a) A barn is 60 ft. long,
40 ft. wide, and 30
ft. high. The slop-
ing edge of the roof
is 24 ft.
(b) Find the area of each
gable end.
(c) Find the total lateral
area.
(d) Find the area of the roof.
4-0'
CHAPTER TEN
SIMILAR FIGURES
A. SIMILAR RECTANGLES
1. Draw four rectangles as follows:
I
//
A. r'xi
B. i"x2"
C. 1" X 2"
D. I"x4"
;
/•
r
Z
B
A-'
C D
2. What is the ratio of the height to the base in each of the
four m?
3. Which ones have the same ratio?
4. Which ones may be considered small maps of another?
6. Draw diagonals in Figs. A and C.
Measure them and find their ratio.
6. Cut out Fig. A and place on Fig. C so that the centers
are together and the diagonals take the same direction.
What position do the sides of Fig. A take compared with
those of Fig. C?
7. Draw a fifth rectangle E^ 1^" by 3". Draw the diag-
onals and place Figs. A^ C, and E together.
142
SIMILAR PARALLELOGRAMS 143
8. Try to place Fig. A with Fig. B or Fig. D.
9. Which of these are aUke in shape?
Figures that are alike in shape are called similar
figures,
B. SIMILAR PARALLELOGRAMS
1. (a) Draw four parallelograms:
A. Sides i" and 1" and included Z 50°
B. Sides 1" and 2" and included Z 80°
C. Sides 1" and 2" and included Z 50°
jD. Sides U" and 3" and included Z 80°
2'
£.
B
3'
2"
(b) What is the ratio of the side to the base in
each O?
(c) Is Fig. B equal to Fig. Cf Why or why not?
(d) Draw the diagonals, cut out Figs. A and B and
try to fit them on each other and the others.
(e) Which ones are similar?
(f) Compare the A in the pairs of similar figures.
(g) In similar figures,
(1) What must be true about their respective A,
taken in pairs?
144 MODERN JUNIOR MATHEMATICS
(2) What must be true about the ratio of their
respective sides?
2. (a) Draw a parallelogram similar to ABCD that is 3
times as large.
7
(b) We sometimes letter two similar figures aUke
except that we put a little accent mark (') be-
side the letters of the second figure. A' is read
A prime.
The O A'B'C'D' is read the parallelogram A
primcy B prime, C prime, D prime,
(c) Z A' is made equal to Z A.
Z fi' is made equal to Z B,
(d) We call Z A and Z A' corresponding angles in
similar figures, or homologous angles.
(e) What side corresponds to ABf to BCf to
CD?
(f) What Z corresponds to Z D?
(g) What is the ratio of A B to A'B\^ of BC to
B'CJ
(h) In similar figures what must be true of all corre-
sponding or homologous angles?
(i) The word similar is used so much that it is con-
venient to have a symbol for it.
The double curve (^) means is (or are) similar
to,
O ABCD ^ O A'B'CD' means the paraUdo-
SIMILAR TRIANGLES 145
gram ABCD is similar to the parallelogram
A prime f B prime, C prime, D prime,
C. SOI^ PRACTICAL USES OF SIMILAR FIGURES
1. All maps, whether of large countries, small sections,
cities, railroad charts, or steamship lines, are figures similar
to the original.
2. All architects' and engineers' plans, whether they be of
a house, a skyscraper, or a railroad bridge, are based on
similar figures.
3. All blue prints of the carpenter or cabinet maker show
him the ideal he is to fashion and give him a working plan.
4. The designer uses similar figures for costumes, wall
paper, carpets, and all kinds of cotton, wool, linen, or silk
cloth as well as for all decorative design.
5. On the outside of a paper dress pattern are small out-
lines of the parts by which they may be identified.
6. The whole process of photography is based on the
idea of similar figures.
D. SIMILAR TRIANGLES
Perhaps the most interesting as well as the most important
similar figures are similar triangles.
1. (a) Draw two triangles of different sizes whose angles
are 40^ 65°, and 75^
146 MODERN JUNIOR MATHEMATICS
(b) Measure their sides to the nearest tenth of an
inch or tenth of a centimeter.
We find XF = 4 cm. ) ^T _ 4 ^ ^^
z'r = 6 cm. I •• x'r " 6 " -^^ "^
YZ = 3.7 cm. ) JZ _ 37
Y'Z' = 5.6 cm. I • • rZ' " 5.6 " ^
XZ = 2.6 cm. ) XZ _ 2j6
Z'Z' = 3.9 cm. ) • • Z'Z' " 3.9 " '^^ "^
The measures cannot be exact, for all measures
are only approximates. But the approximate
value of each ratio is .66 + ; that is, each side
of A XYZ is about f of the homologous side of
A x'rz'.
(c) Are the two triangles similar? Why?
(d) Are two triangles similar if only two angles of
one are equal to two homologous angles of the
other? What must be true of the third angles?
c
4-
2. (a) Draw a A ABC whose sides are 2, 3, and 4 cm.
long.
(b) Draw A A'B'C so that the respective sides are
each 1^ times as long as the corresponding sides
of A ABC,
BC
(c) What is the ratio of AB to A'B'f of ^7^,? of
AC „
A'C
(d) Compare the homologous A of the two A.
(e) What kind of A are ABC and A'B'C'f
SIMILAR TRIANGLES 147
3. (a) Draw several pairs of triangles in which the cor-
responding sides have the same ratio.
(b) Measure the homologous A of each pair.
(c) When the corresponding sides of two triangles
have the same ratio, what kind of A are they?
(d) What is true about the homologous angles of such
triangles?
4. (a) We say the ratio of 9 to 12 is equal to the ratio
of 3 to 4. Written as an equation, i^r = f •
(b) Such a statement that two ratios are equal is a
proportion. We say the numbers are in pro-
portion or are proportional.
(c) In problem' 2, we found the ratios of the corre-
sponding sides were equal,
AB BC
" A'B'~ B'C
Write two other proportions from these figures.
(d) Fill in the correct words in the blanks:
(1) Two triangles are , if their sides are
proportional.
(2) Two triangles are , if two angles of one
are respectively to two angles of
the other.
5. (a) To find the height of a tree from its shadow.
The tree, the shadow, and the rays of the sun
making the shadow may be considered the
A ABC.
(b) Place a stick BE at the end of the shadow of the
tree or at any other place in the sunshine. The
stick, its shadow, and the sun^s rays make the
ABDE.
(c) What is the size of Z A and Z EBD?
(d) The sun is so very far away from the eai:tk tVsa»t.
its rays are consvdet^A ^^x^^.
MODERN JUNIOR MATHEMATICS
Compare / ABC with Z BDE.
(e) la A ABC ~ A BDEf Why?
(f) The height of the tree (A) corresponds to the
height of the stick (h') and their respective
shadows, s and s' correspond.
■ ^ = 1 ^ = i'
" k' s' ■ s s'
Translate these proportions into Enghsh state-
ments.
With tape measure the stick and the two shadows.
If h' = 6'
s = 20'
and s' = 4.
^{k)'h^
.-. A - 30
The height of the tree = 30 feet.
(g) Thales, a Greek who lived over 600 years B.C., made
a trip to Egypt, While there, ha sstoniahed
SIMILAR TRIANGLES 149
Amasis, the King of Egypt, by finding the height
of the pyramid as we foimd the height of the tree.
(h) For what achievement did Thales sacrifice an ox
to the inunortal gods?
(i) Have your teacher tell you or read for yourselves
the story of Thales' mule with its load of salt
and sponges. Reference, Ball's "Short History
of Mathematics."
6. (a) Measure by their shadows the height of your
school building, and of the telegraph poles and
trees in your vicinity.
(b) If a pole or stick is not convenient, the height of
a boy and his shadow may be used,
7. (a) The height of a pole or tree may be found by using
a large 45° rt. A.
(b) The isosceles A ABC must be held perfectly level.
To be sure it is level, a plumb One may be
fastened at the convev C , "^Sfia. 'CSir L^V^^i. 'va
150
MODERN JUNIOR MATHEMATICS
the eye, back away from the tree until the top
of the tree is just visible.
(c) Have the distance measured from your eye to the
ground and from your toe to the foot of the tree.
A ABC - A ADE. Why?
AD = DE, Why?
(d) What must be added to DE to find the height of
the tree?
8. Use your triangles to find the heights of various
objects in your vicinity.
E. THE USE OF THE QUADRANT AND SEXTANT
I. Drawings from Bettinus
1. In order to measure angles accurately, a surveyor has
a very compUcated and very expensive instrument, called a
transit In primitive times a much simpler instrument, a
sextant or a quadrant, was used.
2. (a) The instrument was called a quadrant if it was a
quarter of a circle, and a sextant if it was a
sixth of a circle.
B
(b) An improved and more complex sextant is still
used on shipboard for determining latitude.
USE OF QUADRANT 151
(c) In its simplest form, that used by Thales, a quad-
rant is a frame holding a 90° arc, with a moving
arm. It is used for measuring angles.
Figures A and B show two different styles of
quadrants.
3. To find the width of a river, with a tape and quadrant
/^
^^&&^msM^^^^^r
m^,kw^
* j.f ••*
(a) Let AB and CD be the banks of the river.
(b) Locate some object as A on the opposite bank.
(c) Let one pupil stand at C, directly opposite this
landmark.
(d) Let another pupil walk in the direction of the
line AC to some point E.
(e) With a quadrant, sight an angle of 60°.
Walk along this line until the point F is xi^-s^r.^^
where the Z AFE = Z. E.
MODERN JUNIOR MATHEMATICS
(f) What kind of a A is AFEf
(g) What hnes are equal?
(h) Measure EF and EC to find AC. How?
. (a) The drawings shown are taken from an old book
by Bettinus, printed early in the seventeentli
century. They illustrate the early use of the
quadrant.
(b) The drawing from Bettinus on page 153 shows
the use of a quadrant in finding the depth
of a well.
Which Une measures the unknown depth?
(c) Find two similar triangles in the figure.
(d) Measure three lines and use in a proportion to
find the depth of the well.
(e) Assume reasonable values for these lines and
compute the depth.
USE OF QUADRANT
6. By mounting a quadrant or protractor on a frame a
very good substitute for a transit may be made. This will
hv more servicc!i))Ie if one is mounted in a horizontal posi-
tion and another in a vertical position.
154
MODERN JUNIOR MATHEMATICS
Any pupil who is clever with his hands can make such a
substitute for a transit.
6. (a) Suppose A is a hostile camp, a ship at sea, an
island, or other inaccessible spot.
(b) A gunner at B wants to know the distance AB.
He makes out a line BC A. AB, and a hne
CD _L BC. These lines may be any length.
(c) From D, he sights to A and notes the intersection
with BC at E.
(d) By measuring BE, EC, and CD, he can find the
distance AB, How and why?
(e) Find A5 if BS = 24'
EC ^ 3'
and CD = 40'.
7. (a) Tiie distance may be ioundm aiioXJcvet ^?)u^.
USE OF QUADRANT
(b) Suppose AB is the unknown distance in the lower
figure.
(c) Prolong AB to any point C.
(d) Draw any line from C as CE.
(e) Draw DE \\ BC.
(This may be done by making Z CDE = Z C.)
(f) Sight from E to A. Mark the intersection F.
(g) What & are similar? Why?
(h) Measure BC, CF, FD, and DE. Find AC and
then AB.
(i) Note: This problem is t^en from a book written in
I^atin and printed in 1S45. It is the same book from
which the other drawings were taken.
8. By using one of these methods, find the width of a
river, creek, or street in your vicinity.
9. (a) In the measurements thus far, only one end of the
line has been inaccessible. There is a very easy
method for measurina distMiiea, V«*\\. t"^^ *^
MODERN JUNIOR MATHEMATICS
which are inaccessible. It is by means of un
old instrument called a baculus, meaning rod.
(b) The baculua is really made of two rods, one very
much longer than the other and marked off in
segments of equal length. The shorter rod is
equal in lenf^th to one of the segments and is
made to slide over the longer one easily, but
always remains perpendicular to it.
10. The following picture and explanation showing the
ise of the baculus are also taken from the book byBettinus.
(a) FG is the distance required.
(b) Let CD be at a certain mark on the baculus AB.
(c) Sight from A so that points F and G are just
seen along C and D.
USE OF QUADRANT
157
(d) Then move the shorter rod nearer A, if you ap-
proach FG; or nearer B, if you move away from
FG for the next observation.
(e) Find the point V, so that F and G may still
be just visible past the ends of the shorter rod
at and P.
(f) By measuring the distance AV, between the two
stations, the desired length FG will be had.
Note: The above explanation is tranalated from ttie Latin. The
proof is by proportions derived from several sets of similar triangles,
but it ia too difficult to be given here. The conHtmction and use of the
baculus, however, are very simple.
11. Make estimates of distances between objects on the
opposite bank of a river, then measure with a baculus and
tape.
12- Another way to measure inaccessible distances.
Let AB be the required distance. Standing at C let
observer sight A through D on a rod placed at B.
Do likewise at F, some other convenveTA ■^wo*..
158
MODERN JUNIOR MATHEMATICS
Draw BC, CF, and EF.
Draw BH \\ EF.
It is proved in geometry that if a line is parallel to one
side of a triangle, it divides the other two sides propor-
tionally.
CH BC
HF ~ AB
Therefore,
Which three lines can be measured to find AB?
13. To measure an inaccessible distance by a quadrant vnth
a plumb line, by dravxing a smaU similaT triangle.
Let AB be the required distance.
What kind of a triangle is ABCf
With the quadrant measure angle DCE.
Draw a small right triangle with ZF = ZC.
^,^ FG AC
^^^"^ GH = AB-
Which three lines can be measured to find the dia-
tanceAB?
USE OF QUADRANT 159
The name of this book by Mariua Bettinus is,
Apiaria Universae Pkilosophicae
Mathematicae
Progymnasma Primum
The first problem is called Proposition I. An exact
translation of it is given below.
"Proposition I. To measure an inaccessible distance by
the twenty-sixth proposition of Book I of Euclid.
"Method used by Thalea to measure distances of ships
" I>et A be the position of a ship at sea and let Thales be
on the shore at B. How shall he find the distance ABf
■'Let him withdraw in a straight Une AB to any desired
point, as C, At B, with the aid of a norma, make a perpen-
dicular and mark off any length, as BD.
"Then let the angle £D^ be noted. On the other side, let
the angle BDC be marked off equal to the angle BDA.
'1 assert (Thales says) that if you measm^ the distance
BC, you will know the desired distance AB.
160 MODERN JUNIOR MATHEMATICS
<<
Scholium to Proposition I.
"7/ the point B is on uneven ground , measure off any dis-
tance BF, draw FG perpendicular to AC and operate from G.
'' Subtract BF from H to get poirU C^
Note : A norma is another instrument used by the ancients to draw
perpendicular hnes. It consists of three rods placed at right angles to
each other.
Note the lack of perspective in the drawing.
If the angles BDA and BDC are equal, what kind of a
triangle is ADCf
Why does BC equal AD?
Bettinus gave a proof for this proposition by proving the
triangles ABD and CBD equal.
CHAPTER ELEVEN
POLYGONS
A. TRAPEZOIDS
1. (a) Roads and streets do not always run at right
angles to each other, but lot lines are usually
perpendicular to the street on which the lots
face. In such a city block most of the lots are
rectangular, but a few will be in the shape of
triangles and trapezoids.
(b) A trapezoid is a figure inclosed by four straight
Unes, only two of which are parallel.
(c) The two parallel sides are the bases. The two
non-parallel sides are the legs.
(d) If the two legs are equal, the figure is an isosceles
trapezoid.
Trapezoid
Isosceles Trapezoid
/z
/
/^
-f
5
6
Vq\
162 MODERN JUNIOR MATHEMATICS
2. (a) What is the shape of each lot in the above plot?
(b) In order to find the area of lots 6 and 12, one
must know how to find the area of a trapezoid.
(c) The symbol for trapezoid is o.
3. (a) Draw a trapezoid ABCD and its altitude h.
(b) Draw one diagonal as BD,
(c) Into what two parts does BD divide the trapezoid?
(d) We see that A 1 + A 2 = O ABCD.
(e) Let 6i (read b sub 1) = the base of A 1 or the
lower base of the O; and 62 (read b sub 2) =
the base of A 2 or the upper base of the O.
(f) What is the height of each A? The height of
the trapezoid is a line perpendicular to the bases.
(1) Sai =^bih
(2) >Sa2 = ^ b2h
(3) S^ = i 61/1 + i 62/1
(4) Take out the common factor | h.
(5) /. /Sc, = n (bi + &2).
(g) Translate this formula into an English statement.
4. If 61 = 10 cm., 62 = 5 cm., and A = 3 cm., find Sex-
Solution.
Then Sc^ = i /i (61 + 62)
= i X 3 (10 + 5)
= i X 3 X 15
— 45
- IT
= 22^ sq. cm.
TRAPEZOIDS
163
6. The fonnula for the area of a trapezoid may be found
in a different way.
6.
(a) Draw two trapezoids that are exactly equal.
(b) Cut out one and place it beside the other as in
the illustration.
(c) What kind of a figure is the result?
(d) What is the base of the new figure? The height?
(e) What is its area?
(f) What part of the new figure is the trapezoid?
(g) Therefore, what is the area of the trapezoid?
(a) Find the areas of the following trapezoids.
(b) Use the formula
No.
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
61
10"
8"
2' 6"
140'
13i'
a + 3
2x + 7
a4-26
2c4-3d
18.3'
x-\-y
165.7'
5"
6"
1'6"
50'
8i'
a4-2
2a; + 3
a + 6
c-\-2d
10.5'
X
95.1'
h
4"
7"
9"
\
96'
5r
4a
Sx
2a
2c
8.2'
2x
Via'
s
\
164
MODERN JUNIOR MATHEMATICS
I
I
I
X,
TUX
7. In the diagram page 161 measure lots 1, 6, 7, 8,
and 12 to the nearest tenth of a centimeter. Find their
areas if 1 cm. equals 60 ft.
8. (a) A room has a bay window and is shaped like the
diagram.
(b) The length of the room is
24 feet.
(c) Find the scale and the
entire area of the room
including the bay-
window.
9. (a) Draw a trapezoid with a
very small upper base.
What other figure does it resemble?
(b) Show that the formula for the area of a trapezoid
may be used for the triangle by considering
the upper base 0.
10. (a) To measure an irregular shaped piece of land, as
that in the bend of a river, a straight line may
be run so as to cut off the bend, as AB. At
equal intervals on AB perpendiculars are run.
These divide the land mto approximate trape-
zoids.
w j24' ^
(b) This figure is drawn to a scale of 1" to 40'.
Find the length of parts AB and of the perpen-
diculars to AB,
(c) Find the area of each neai tiapeTiovd.
POLYGONS 165
(d) Find the total area of the strip of land in the
bend.
(e) The perpendicular lines are called offsets.
Find the sum of the offsets and multiply it by
conamon altitude AX. How does this result
compare with your total area? Show why
this method may be used if the offsets at the
ends A and B are 0?
B. OTHER POLYGONS
1. Any figure bounded by three or more straight lines
is a polygon.
The word polygon means many sided.
What special kinds of polygons have you used?
2. Polygons have special names according to the num-
bers of their sides.
(a) A triangle is a polygon with 3 sides.
(b) A quadrilateral is a polygon with 4 sides.
(c) A pentagon is a polygon with 5 sides.
(d) A hexagon is a polygon with 6 sides.
(e) An octagon is a polygon with 8 sides.
(f) Other polygons have special names, but these are
the most common. They are used extensively
in design, especially for tile, Unoleum, and wall
paper patterns.
3. (a) If all the sides of a polygon are equal and all of
its angles are equal, it is a regular polygon.
(b) What kind of a triangle is regular?
(c) What kind of a quadrilateral is regular?
(d) Is a rhombus a regular polygon? Why?
(e) Name some special kinds of triangles.
(f) The word polygon alone means an irregular one.
A regular polygon is sp^e\a.\\^ tqkv^Ssss^^^^
166
MODERN JUNIOR MATHEMATICS
(g) The following figures show the regular and ir-
regular polygons of five, six, and eight sides.
Regular Pentagon
Pentagon
REoniiAR Hexagon
Hexagon
Regular Octagon
Octagon
REGULAR POLYGONS 167
C. CONSTRUCTION OF REGULAR POLYGONS
A polygon is inscribed in a circle when it is drawn inside
the circle so that each vertex Ues on the circumference.
The word inscribe literally means M write in or to
draw in. The regular hexagon is the easiest to construct,
therefore we shall learn that first.
L The Hexagon
1. To construct a regular hexagon.
(a) Draw a circle with the desired radius.
(b) Draw any diameter, as AB.
(c) With A and B as centers and with the original
radius draw arcs cutting the circle on each
side of A and B.
(d) Join the points of intersection in succession to
make a regular hexagon.
(c) Note: In making constructions, do not lift the compass
point until all possible arcs from that center are made.
2. (a) Make another regular hexagon of a different size..
(b) Measure the Z at eac\LNet\,eYLm\ic>N^^^e;iX5^.
168 MODERN JUNIOR MATHEMATICS
(c) Draw all the radii. What shaped figures result?
Measure their sides.
(d) How many degrees are in each Z of each A?
3. (a) Make a third regular hexagon.
(b) Join three alternate vertices with straight lines.
(c) What kind of a figure results?
4. (a) Draw another regular hexagon, a rather large
one with a radius of about 1^".
(b) Draw all the radii.
(c) Measure each Z at the center.
(d) Bisect each Z at the center by other radii.
(e) Join each of these new points on the circumference
to the two nearest vertices of the hexagon.
(f) How many sides has the new figure? Are they
equal?
(g) In Latin dvx) means two, decern means ten, and
duodecim means two plus ten or twelve. Similarly,
in Greek the word for twelve is dodeka and for
angle is gonia. Therefore this new twelve-
sided polygon is called dodecagon.
6. (a) Draw a regular hexagon and an irregular one.
(b) Measure the angles at the vertices in each.
(c) Find the sum of these angles in each hexagon.
(d) Change the measure of these sums in degrees to a
measure in right A,
How do the two sums compare?
6. (a) How many regular hexagonal tiles will exactly
fit together at one point?
(b) How many square tiles?
(c) How many regular triangular tiles?
7. Draw a regular hexagon whose side is 2"; If".
8. (a) Draw a regular hexagon and three radii to alter-
nate vertices.
(bj What shaped figures result?
REGULAR POLYGONS 169
(c) Join the ends of these radii.
(d) How do the resulting triangles compare?
(e) How Boany of these triangles are in the hexagon?
(f) How many are in the inscribed triangle?
(g) What is the ratio ol tt« tma'^ "wj '0wb\«sMM*5s^
170 MODERN JUNIOR MATHEMATICS
9. (a) With each vertex as a center and with the original
radius draw arcs within the hexagon to make a
conventional flower.
(b) Draw these only at alternate vertices.
10. By combining different lines and arcs within a regular
hexagon, and shading some of the parts, different decorative
designs may be made.
11. (a) Draw a regular hexagon in a circle.
(b) Join three alternate vertices to make a regular
triangle.
(c) With each vertex of the triangle as a center and
its side as a radius, draw an arc.
(d) Shade or color different parts.
12. (a) Draw a regular hexagon.
(b) Bisect each side. (Bisect one side and use its
half as a measure.)
(c) Join the mid-points of the sides in succession.
(d) What kind of a figure is formed?
13. (a) Draw a regular hexagon.
(b) Join the six pairs of alternate vertices.
(c) How many points are in the resulting star?
(d) What is the shape of the figure left in the center?
14. To make a regular hexagon from an equilateral triangle,
(a) Draw and cut out an equilateral triangle about
10 cm. on a side.
(b) Either by folding or drawing two altitudes, find
the center of the triangle, that is, the intersec-
tion of the altitudes.
(c) Fold over each comer of the triangle, so that the
vertex just touches the center.
(d) Measure the sides of the resulting hexagon.
REGULAR POLYGONS
171
IL The Octagon
The construction of the octagon is based upon the square.
1. To construct a regular octagon,
(a) Draw a circle with
two ± diameters.
(b) Bisect each angle at
the center 0.
Only one Z need be
bisected, as ZAOC,
for an ^^ equal to
AB may be laid off
on each quarter.
(c) Join the eight points
on the circum-
ference in succes-
sion to make a reg-
ular octa^xN..
172 MODERN JUNIOR MATHEMATICS
2. (a) Make another regular octagon of different size.
(b) Measure the Z at each vertex in both figures.
How many degrees in each?
How many right A ?
(c) Draw all the radii.
(d) How many and what kind of A are formed?
(e) How many degrees are in each Z of each A?
3. What would you do to an octagon to make a 16-sided
polygon?
4. (a) Draw a regular and an irregular octagon.
(b) Measure the angles at the vertices in each.
(c) Find the sum of these angles in each octagon.
(d) What does the sum measured in degrees equal
when measured in right angles?
(e) How do the two sums compare?
(f) Does the sum of the interior angles of an octagon
change with its size or shape?
6. (a) Can you put three regular octagonal tiles together
so that the comers exactly fit?
(b) If you put two together, will there be any space
left?
(c) What shaped figure will exactly fit in the space
left?
6. (a) Join the four alternate vertices of a regular octa-
gon in succession.
(b) What shaped figure i^ formed?
(c) Join the other four pairs of alternate vertices.
(d) What shaped figure is left in the center?
7. (a) Draw several octagons in circles.
(b) By joining alternate vertices, drawing diameters
and arcs of circles and shading or coloring dif-
ferent parts, make different designs.
REGULAR POLYGONS 173
174
MODERN JUNIOR MATHEMATICS
8. To make a regular octagon from a square by folding.
(a) Draw and cut out a square 10 cm. long.
(b) By folding bisect each side of the square.
(c) Fold over the comers, making an inner square
EFGH,
(d) Open the paper and then fold over the comer A
so that AE falls along EH,
(e) Open again and fold over the same comer so that
AH falls along EH.
D G C
H ri-
A E B
(f) These folds will intersect, as at X.
(g) In like manner fold over each comer,
(h) The resulting figure is a regular octagon.
m. The Pentagon
1. To construct a regular pentagon.
The greatest care must be used and pencils must be very-
sharp in the construction of a pentagon. If the compass
point is off by the width of a pencil line, the pentagon will
not be exact.
(a) Construct a circle and divide the diameter AB
into five equal parts.
(b) Find C, the vertex of an equilateral A on AB.
(The A need not be dtawiv.^
REGULAR POLYGONS
c
175
(c) Through £, the second mark on the diameter,
draw a line from C to meet the circmnference
atD.
(d) Draw AD,
(e) Use AD as a radius and mark off four other equal
arcs on the circumference.
(f) Join these points in succession to form a regular
pentagon.
2. (a) Make another regular pentagon.
(b) Measure the Z at each vertex in both figures.
How many degrees are in each Z ? How many
right A ?
(c) Draw all the radii.
(d) How many and what kind of A are formed?
(e) How many degrees are in each Z of each A?
3. What would you do to the pentagon to make a regu-
lar 10-sided polygon, that is, a decagon?
4. (a) Draw an irregular pentagon.
(b) Measure the A at the vertices and find their
sum, measured in degrees and iiv rv^t. A.
176
MODERN JUNIOR MATHEMATICS
(c) How does the sum of the A in this one compare
with the sum of the 2^ of a regular pentagon?
(d) Does the sum of the interior 2^ of a pentagon
change with its shape or size?
5. Does the sum of the interior 2^ of a polygon change
with the number of sides?
6. Make a table showing the sum of the interior A of
the different polygons constructed and the size of the Z
at each vertex, measured in rt. 2^ and in degrees.
Angles of Regular Polygons
Name of
Polygon
Number of
Sides
Sum of the
Interior A
No. of Rt. A
in each Z
No. of Degrees
in each Z
Triangle
Square
Pentagon
Hexagon
Octagon
3
4
5
6
8
2rt. A
frt. Z
60°
7. (a) Draw a regular pentagon and all possible diagonals.
(b) How many points are in the star?
(c) What kind of A are formed on each side of the
pentagon?
REGULAR POLYGONS 177
(d) What kind of a polygon b left in the center?
(e) Compare its position with the original pentagon.
(f) There are now three A at each vertex.
Compare them in size,
(g) How many isosceles A can you find in the ^ure?
(h) Without measuring, can you compute the size of
each of the four A at one vertex of the inner
pentagon?
8. Can you find a rhombus in the figure? How many?
9. (a) Can you fit four regular pentagonal tiles together?
(b) Can you fit three together?
(c) How much of an angle will be left?
(d) Is there any regular polygon that will fit in exactly?
10. Make and shade or color several ordinal designs
made from pentagons.
178 MODERN JUNIOR MATHEMATICS
11. It is impossible to construct exact regular polygons
of 7, 9, or 11 sides with compass and ruler. But the method
given for the pentagon, which gives approximate results
only, may be used for the other polygons.
To construct a heptagon or 7-sided polygon, divide the
diameter into 7 equal parts, and from the vertex of the
equilateral triangle draw a line through the second point of
division. Proceed as in the construction of the pentagon.
CHAPTER TWELVE
CYLINDERS AND CIRCLES
A. CYLINDERS
1. We have seen that many objects in our surroundings
are more or less rectangular. But there is another group
of objects about us that have one general shape different
from the rectangular solid.
2. Think for a moment of the shape of a telegraph pole,
the trunk of a tree, the water glass for the table, the ice
cream freezer, the fruit jar, the rows of tin cans containing
vegetables on the grocer's shelves, smoke stacks on steam-
ships, the pencil with which you write,
the pillars in front of some pubHc
buildings. All of these things are cylin-
drical in shape, or like a cylinder.
3. The tin can is probably the most
convenient example of a cylinder.
(a) How many surfaces has it?
(b) How many of these are flat
or plane?
(c) How many are curved?
(d) What is the shape of the flat
surfaces?
(e) What is their relative posi-
tions?
4. (a) To test a plane surface, put the edge of your
ruler on it in several different positions. If the
edge touches the surface in all positions, it is a
flat or plane surface,
(b) Put the edge of your ruler along the side of a tin
can or other cylindrical object, itftwi. ^-^ "va
180
MODERN JUNIOR MATHEMATICS
bottom. Does the edge touch the surface at all
points?
(c) Put the edge in several other positions and see if it
touches.
(d) If the edge does not touch the surface in all posi-
tions, the surface is curved.
(e) There are many kinds of curved surfaces.
(1) A ball has one kind, called a spherical surface.
(2) The side of the tin can is another, called a
cylindrical surface,
(f) Revolve a rectangular card about one edge. The
opposite edge traces a cylindrical surface and the
whole card traces a cylinder.
5. The only kind of a cylinder we shall consider is one
in which the side is perpendicular to the base.
6. A cylinder is a figure enclosed by a cylindrical surface
and two parallel circular bases.
CIRCLES 181
7. (a) Imagine a tin can cut straight down the side and
the top and bottom ahnost cut off and the tin
pressed out flat.
An illustration of this pattern is shown on page 180.
(b) The cylindrical surface is called the lateral surface
of the cylinder. What is the shape of the lateral
surface in the pattern?
(c) What is the height of this rectangle?
What is its base?
(d) We see, therefore, in order to find out how much
tin it will take to make this can, or how big to
make the label for it, we must know how to
find the circmnference of a circle.
B. CIRCLES
I. Circumference of a Circle
1. (a) Carefully measure aroimd the can and measure its
diameter.
(b) How many times larger than the diameter is the
circumference?
(c) Measure a glass and several other cylinders in the
same way and find the ratio of the circumfer-
ence to the diameter in each case.
(d) Is the ratio the same for all circles?
(e) We cannot measure this ratio exactly, but it is
nearly 3^. That is, every circumference is
about 3^ times its diameter, or, as a decimal,
3.1416 times.
(f) This ratio is used a great deal by mathematicians
and for convenience they use as a symbol for
it the Greek letter for p, called Pi. The symbol
is TT.
182 MODERN JUNIOR MATHEMATICS
(g) The circumference of a circle is 3^^ times its di-
ameter.
Let c = circumference
TT = 3| or 3.1416
and d = diameter.
Then the formula is
c =7rd.
(h) Since the diameter is two times the radius, we may
put 2 r in place of d, and have
c = 27rr.
Translate this formula into an English statement.
2. For the cylinder
Lat. S =^ ch = 2t rh or IT hd.
Translate into an English statement.
3. A coffee can has a 4^" diameter and is 5J" high.
How much paper will be needed to furnish labels for 5000
cans?
4. (a) A farmer's silo is 12 feet in diameter and 40 feet
high. How many square feet are in its lateral
area?
(b) How many gallons of paint are needed to paint
the outside, allowing one gallon to 250 sq. ft.?
(c) If a painter can paint one square an hour and
receives $.65 per hour, how much does he re-
ceive for the painting?
(d) What is the total cost to the farmer?
5. A girl is buying a stamped doily to embroider. At
the same time she wants to buy the lace edge for it. The
circular doily is stamped on a square of Unen. In one
comer is stamped the diameter of the doily.
(a) How much lace 1" wide will be needed for a 22"
doily?
(1) What will be the diameter of the firiis.H'jd
doily after the lace is aewed on?
(2) In order to have the lace he flat, the exact
amount plus J" to f" for the seam is
needed.
(3) No allowance need be made for "f ullin g on,"
because the circumference of the doily itself
is enough smaller than the edge of the
lace to allow for fulhng.
(b) What will the lace cost at $.18 a yd.?
6. A circular lunch cloth just covers the top of a 54"
round table.
(a) How much lace 3J" wide will be needed for the
edge of the cloth?
(b) What will the lace cost @ $.37| a yard?
7. A luncheon set contains one 24" centerpiece, four 7"
plate doihes, and eight 3" tumbler doihes.
The centerpiece has lace 1^" wide @ $.35 a yard; the
plate doilies have lace I" wide @ $.25 a yard; and the
tumbler doilies have lace Y' wide at $.18 a yard.
(a) How much lace ol eac\Y^n.^'\s'QBJi&«^
i4 MODERN JUNIOR MATHEMATICS
(b) Find the cost of each kind and the total coet of
the lace.
8. (a) The wheels of one automobile have 32" tires.
How many revolutions will the wheels have to
make in going one mile?
(b) The wheels of another machine have 36" tires.
Which wheels make more revolutions per mile,
and how many more?
(c) Such calculations have to be made in setting
speedometers, for a speedometer set for one
sized tire will not register correctly if a different
sized tire is put on the machine.
(d) Find the circumference and the number of revo-
lutions per mile for a 33" tire; for a 38" tire.
9, A circular flower bed is ten feet in diameter.
(a) How much wire fencing is needed for it?
(b) Two or more circles
are concentric if
they are drawn
about the same
center with dif-
ferent radii.
Con-centric means
centers together.
(c) The owner of the
flower bed wants
to fill it with aster
plants, putting one
at the center and
the others in concentric circles 6" apart. In each
circle the plants are 6" apart. The outermost
circle is within 6" of the fence.
(d) Hovf many plants must \« 'pvYnAvaseAl
CIRCLES 185
n. Area of Circle
We have found how to measure the lateral area of a
cyUnder. But in order to find how much tin it will take
to make the can we must know how to find the area of the
top and bottom; i.e., to find the area of a circle.
1. To find the area of a circle,
(a) Draw and cut out a circle.
(b) Fold over on its diameter.
(c) Fold again and again imtil it looks like a very
small piece of pie.
(d) Unfold and count the number of parts. There
should be at least sixteen.
(e) Cut through the circumference to the center in
just one place.
(f) Cut from the center along each fold almost to the
circumference.
(g) The circle has been cut into a series of parts that
are very much Uke triangles.
(h) The height of each triangle is the radius of the
circle and the sum of the bases of the A is the
circumference of the circle, or 2 tt r.
SoiA^^bh
Substituting 2 tt r for 6, and r for ft, we get
1
2
'So= rt X iirr X r
So = 7rr^.
» MODERN JUNIOR MATHEMATICS
(i) Translate this formula for the area of a circle
into an English statement.
2. (a) Draw a circle.
(b) Draw a square on one radius.
(e) What is the area of this
square?
(d) How many times larger
than the square is the
circle?
(e) The formula S^-ttt^
means that the circle
is about 34^ times a-s
lai^ as a square dran-n
on the radius.
. Find the area of a circle 7" in diameter.
Solution.
Formulas© =n'r^
Given d - 7'
Then r -l
5a = 3= X ^
-^ 7 7
77
- -5- " 38i sq. in.
(a) Instead of squaring J and using *^, the square
was indicated as ^ x ^. This plan often saves
much computation. In this problem two can-
cellations could be made.
(b) It is better not to perform a multipHcation or divi-
sion until it becomes necesawrj .
CYLINDERS 187
C. SURFACE OF A CYLINDER
1. Find the number of square inches of tin in a can 4^
in diameter and 5§" high.
//
Solution.
9
(a) Given d = 4^ or j:
r = T
r2 =
9
4
9 9
4X4
2 bases = 2 x tt r*
11
(b) Given r
h
« 22 9 9
=2Xy XgXj
2
891
~ 28
= 31.8 sq. in.
9
4
^ 11
2
188 MODERN JUNIOR MATHEMATICS
Lat. S = 2Trrh
11
^ 3er 9 11
1089
" 14
= 77.8 sq. in.
Lat. S = 77.8
2 bases = 31.8
Total S ^ 109.6 sq. in.
Note: One must exercise one's judgment as to the advisability of
using the 3} or 3.14 or 3.1416 as the value of x. It would be absurd
to calculate the area of a tin can to four decimal places. Even finding
it correct to one tenth of a square inch is impractical, for there must
be some allowance for waste and seams.
D. fflSTORY OF PI
It was not until the middle of the eighteenth century
that the Greek letter ir came into use as a symbol for the
ratio of the circumference to the diameter of a circle. But
from earUest times mathematicians knew that there was
such a constant ratio. Different values were given it in
different periods of history.
1. One of the earliest books we have is the Ahmes
(Ah'mez) Papyrus, written about 1700 b.c. In this manu-
script a value is given to ir equal to ^^^ or 3.1604.
2. The Jews and Babylonians considered tt equal to 3.
This fact is shown in the measures given for sacred vessels
in I Kings vii, 23 and II Chronicles iv, 2.
3. Archimedes of Syracuse, who lived between 287 and
212 B.C., was a great mechanical genius as well as mathe-
matician. You will find it interesting to read the stories of
his detection of the fraudulent goldsmith; his use of burning
HISTORY OF PI 189
glasses to destroy the Roman ships; his apparatus for
launching ships; and the Archimedean screw used to drain
the flooded fields of Egypt.
Archimedes proved that the value of tt is between S^J-
and 3|f .
We can understand these values better by putting them
in decimal form, but Archimedes did not have this advan-
tage, because no one knew anything about decimal fractions
until nearly 1600 a.d.
3| = 3.1428
7r= 3.14159 +
3ff = 3.1408. ^
«
4. Ptolemy, a great astronomer of Alexandria about
150 A.D., used S^jj as the value of tt. As a decimal
3xVo = 3.14166.
5. Between_400 and 600 a.d. the Hindus used tt = 3 or
3 J and tt = VlO which is 3.1622^
The Chinese had used tt = VlO about 200 a.d.
6. The exact value of tt cannot be expressed in ordinary
figures, although many persons have contended long and
earnestly that it could be done. If this were possible, a
square could be constructed exactly equal to a circle. These
people are known as ^^circle-squarers."
About 1600 the value of tt was calculated to 35 decimal
places. Since then it has been calculated to 707 decimal
places, but it will never come out ''even." In other
words no square can be constructed that is exactly equal
to a circle.
The value correct to the first 35 places is as follows:
TT = 3.141592653589793238462^383279|^88
When very exact measures are needed, we use tt =
3.14159 or 3.1416.
For less exact measures, we \iseir = ?>.W c^^'tc = "^^^
MODERN JUNIOR MATHEMATICS
E. PROBLEMS — CYLINDERS AND CIRCLES
1. Find the areas of the following circleB :
(a) r = 3J" (d) J- =1 ft. 8 in.
(b) d = 5 ft. (e) d = 22"
(c) r = 6| cm. (f) r - 150 ft.
2. Find the formula for the area of a circle in terms of
the diameter instead of the radius.
3. (a) Most modern houses and buildings are heated b}'
passing steam or hot water through
a set of cylindrical pipes called a
radiator. The surface of these pipes
is called the radiating surface, and
its size depends upon the size of' the
room. It is measured in square feet.
(b) Find the amount of radiating surface
from 12 pipes, 2" in diameter and
35" h^h. (Disregard the bases.)
(c) A radiator has two rows of ten 2" pipes 32" high.
How many square feet of radiating surface has it?
(d) How much larger room will the second radiator
heat than the first?
. (a) The amount of water
that flows through
a pipe depends
upon the area of its
cross section, that
is, upon the area
of the circle inside the pipe.
(b) Find the area of a cross section of a pipe whose
inner diameter is 8".
(c) Thethicknessof the iron of this pipe is 1", What is
the diameter of the outer cross section?
Find its area.
CYLINDERS AND CIRCLES
191
(d) From these two areas how may the area of the
ring be found?
6. There is an easier way to find the area of a ring.
Let the outer circle be O 1 with a radius of ri or 5, and
the inner circle be O 2 with a radius of r^ or 4.
O 1 = TT ri^
O 2 = TT r22
By factoring
By substitution
Ring = O 1 - O 2 =7rri2 -irr^^
= TT (j^ - r^)
= TT (52 - 42)
= TT (25 - 16)
= 7r9
= 3.1416 X 9
= 28.2744 sq. in.
6. (a) The radius of the inner cross section of the first
pipe is 3 in. and of the second 2 in. How
many times larger is the first than the
second?
192 MODERN JUNIOR MATHEMATICS
(b)
SolvMon.
In O 1, ri
= 3
In O 2, r2
= 2
Area of O 1
= 7rri*
Area of O 2
= 7rr2*
The ratio of these areas
7rri2
Trr2
by reducing to lowest terms
3''
= ^or2i
/. O 1 is 2J times as large as O 2.
(c) Do you have to compute the exact areas of two
circles to find the ratio of their areas?
(d) The areas of two circles have the same ratio as
the squares of their radii or as the squares of
their diameters.
Show why radii or diameters may be used in these
ratios.
7. (a) The rate of the flow of water through a cylindrical
pipe is proportional to the area of its cross
section,
(b) Two pipes have 1" and 2" inside measurements,
respectively.
_ 1
~ 4
Therefore 4 times as much water will flow per
minute through the second pipe as through the
first.
8. How much more water will flow per minute through a
3'' pipe than through a 1" pipe?
CYLINDERS AND CIRCLES 193
9. How much faster will water flow through a 4" pipe
than through a 1|" pipe?
10. In a park is a fountain in the center of a circular grass
plot 300 ft. in diameter. A 10-ft. cement walk surrounds
the plot. What is the area of the walk?
11. (a) If the equatorial diameter of the earth is 7924
miles, how big is the equator?
(b) How many miles long is V at the equator?
12. (a) A company advertises for bids for painting 9-ft.
bands around telephone poles whose average
diameter is 14".
(b) Mr. A figures paint at $3.00 per gallon and allows
1 gallon to 275 sq. ft. He figures 1 hour's
time for painting a square and 7 hours' extra
time per C poles for moving material from one
to another. The labor costs $.70 per hour.
(c) Mr. B makes a bid of $65 per C poles, with all
materials furnished.
(d) To which man should the company give the con-
tract? How much is saved thereby?
13. (a) An oatmeal box is 7" high and has a diameter of
4j". How large must be the paper used for
the label around it?
(b) How much cardboard is needed for the box?
14. (a) A box of Dutch Cleanser is 4f " high. Its diam-
eter is 3-'e How large is the label covering
the side?
(b) How large is the tin in each end?
CHAPTER THIRTEEN
VOLUME
A. VOLUME OF A PRISM
1. A cube 1 cm. long is a cubic centimeter.
2. A cube 1" long is a cubic inch.
A cube 1' long is a cubic foot.
3. (a) How many 1" cubes can be laid
on the bottom of a cubical
box 1 ft. long?
(b) It is evident that there can be
one cube for each square inch
of surface of the bottom.
(c) Since 1' = 12", there can be
12 X 12 or 144 cubes in one
layer.
(d) Since the cube is 12" high,
how many layers of cubes
can be put in?
(e) Evidently the total number of
small cubes in the box is 12
times 144 or 1728.
Therefore 1 cu. ft. = 12^ = 1728
cu. in.
4. Suppose the box were only 8" high. The number of
cubes would be 8 X 12 x 12.
6. Suppose the box were 9" wide. Then each layer
would have 12 x 9 cubes.
6. If the box were 12" long, 9" wide, and 8" high, the
number of cubes would be 12 x 9 x 8. Such a box is a
rectangular prism.
7. The number of cubic inches the box can contain is
called its volume,
194
VOLUME
195
Volume of other boxes may be measured in cubic centi-
meters, cubic feet, or cubic yards.
8. If I = length
w = width
and h = height of a rectangular solid, and V = volume,
then V = Iwh.
9. If the base of this box were a tri-
angle, one cube could be placed on each
square inch of the base, and the volume
would be the area of the base x height.
V = Bh
10. The same statement will hold true
if the base is a hexagon, other polygon, or
a circle.
Triangular Prism
J
\
Hexagonal Prism
Cylinder
B. VOLUME OF A CYLINDER
1. (a) In each of these solids the area of the base, B,
will have to be computed separately according
to its shape.
(b) For a cylinder, we know the area of the base is tti^.
Therefore wr^ may be substituted for Bj giving
V cyl. = IT r^ K.
196 MODERN JUNIOR MATHEMATICS
2. (a) To find the area of any surface, both dimensions
must be measured in the same imit of length.
To find the volume, the three dimensions must
be in the same unit of length,
(b) How many dimensions must be known to find
the volume of a cylinder? What are they?
C. PROBLEMS
1. (a) How many cu. ft. of water will a tank hold that
is 8' X 6' X 5' inside measurement?
When dimensions are given as above, it is under-
stood that they are given in the order of I, w, and h,
(b) How many gallons of water are in the tank when
full? Allow 7| gallons to the cubic foot.
(c) How many gallons are in the tank when the
water is 2 feet deep?
(Be sure you use the shortest method for your
computation. Compare with 1 (a)).
(d) What is the weight of the water in the tank when
full? One cubic foot of water weighs 62.5 lb.,
or in other words, the density of water is 62.5
lb. per cu. ft.
Note: We have seen that it often pays not to perform a
multiplication or division mitil it is absolutely neces-
sary. Take such a problem as to find the number of
gallons in a container 16" x 12" x 15". This problem
may be solved
(1) By finding the number of cu. in., dividing by 1728,
and multiplying by 7J.
(2) By reducing each dimension in inches to feet, multi-
plying together and by 7J.
(3) By finding the volume in cu. in. and dividing by
231, the exact number of cu. in. in one gallon.
Allowing 7^ gallons to the cu. ft. is the same as using
230 cu. in. to the gallon.
(4) Or, the plan of No. (1) may be used, but each process
indicated and then all performed at one time.
VOLUME
197
For comparison the two plans are given below.
Plan No. (1)
y = 16 X 12 X 15
= 2880 cu. in.
2880
2880 cu. in. = rzrr cu. ft.
1728
= If cu. ft.
1 2
1728)2880
1728
1152 288 36
2
1728 ~ 432 " 54 '
"3
No. of gal. = 7i X If
5
}^ 5
= 2^3
25
" 2
= 12i gal.
Plan No. (4)
V = 16 X 12 X 15 cu. in.
16 X 12 X 15
" 12 X 12 X 12 ''''• ^^•
B 5 5
Jt^Txl^xl^ 1^
No. of gal.
i i
2
25
° 2
= 12J gal.
2. A mason jar has a diameter of 3^''. It is 6" high.
Show that it is exactly a quart measure.
3. A half-pint measuring cup is 2f " in diameter and 2\"^
high. Is it exactly a half-pint measure?
4. (a) For proper ventilation the law in most states re-
quires at least 200 cu. ft. of air per pupil in a
schoolroom.
(b) In a schoolroom 36' x 24' x 12', what is the
largest number of pupils that should be en-
rolled?
(c) If the ceiling were two feet lower, what difference
should be made in the enrollment of the room?
6. Measure your schoolroom to see how many pupils it
can safely accommodate.
CHAPTER FOURTEEN
REVIEW OF FORMULAS
A. TRANSLATION OF FORMULAS
1. Pn = 4 e
2. Pn = 2 (Z + w)
3. Sa = e"
4. So == hw ^ bh
6. Lat. ScvL. = 4 e^
6. Tot. ScM, = 6 c2
7. Lat. Sob. = 2h (l + w)
8. Tot. Sob. ^2lw + 2lh + 2wh
9. Sa = I bA
10. Rt. A : c = Va2 + 6^
11. Sc. = ^ A (6i + 62)
12. Co= 27rr
13. S0=7r.r2
14. Lat. Scyi. = 27rrA
16. Vcu. - e^
16. Va pr. = iiy/i
17. 7pr. = M
18. Vcyi. = TT r2/i
B. FORMULAS GIVEN IN THE SUPPLEMENT
1. Heroes formula for the area of a triangle.
S = Vs (s -a)(s -6)(s-c)
2. Lat. opyr. = ^ PI
4. Lat. Sco. = 'TT rZ
6. Fco. = I TT r^ft
6. Ssph. = 4 TT r^
7. Fsph. = 4^r3
1^^
SUPPLEMENTARY TOPICS
A. INSPECTION METHOD OF FINDING SQUARE ROOT
The third method of finding square root is by inspection.
By knowing the squares of numbers to 30, one can find by
inspection the square root of certain numbers correct to
one or two decimal places.
1. Example: Find the square root of 5.
'529 = 232 Diff. 29
VS = V5M) = 2.2 + = 2.23 +
.484 = 222 Diff, 16
Explanation: Annex two decimal ciphers to 5. Disregard
the decimal point for the moment and consider the number
500. The square next higher is 529 or 23^. The one next
lower is 484 or 22^. Then the V5OO must be between 22
and 23, i.e. 22 +. By putting the decimal point in again,
we get VK06 = 2.2 +.
A little practice in estimation will give the next digit.
The difference between 484 and 500 is 16; between 500 and
529 is 29. These show that 500 is less than half way be-
tween the two known squares. Therefore, the next digit in
the root is less than 5. By comparing the difference, one
can estimate the root to be 2.23 +.
Extract the root and compare results.
2. Another example. Vs =?
841 = 292 Diff. 41
Vs = V8.OO = 2.8 + = 2.82 +
784=282 Diff. 16
Extract the root and compare results.
200
MODERN JUNIOR MATHEMATICS
3. By inspection, find the square roots correct to two
decimal places of 2, 3, 6, and 7.
4. Find correct to one decimal place the square roots of
175, 150, 205.
6. A square field contains 10 acres. What is the length
of each side? What would wire fencing for it cost at 90^
per rod?
6. By inspection, find correct to one decimal place the
square roots of:
(a) 80 (e) 56 (i) 135
(b) 180 (f) 45 (j) 700
(c) 32 (g) 108 (k) 535
(d) 105 (h) 600 (1) 1000
B. HERO'S FORMULA FOR FINDING THE AREA OF
A TRIANGLE
1. Sometimes it is easier to measure the three sides of
a triangle than to measure the base and altitude. This is
true of any plot of ground that has some obstruction in the
center, as a house or pond or swamp.
About 100 B.C. a Greek surveyor at Alexandria, Egypt,
found a' way to measure a triangular field from the length
of the three sides. The surveyor's name was Hero and the
formula is known as Hero's formula.
2. If the sides of a triangle are 5", 12'', and
13", he takes half the sum and calls it s, which
is 15. From this half sum he subtracts each
side in succession, getting the remainders 10, 3,
and 2. He multiplies these remainders by the
half sum and extracts the square root of the
product.
s ^ ^ (a-\-b + c)
= H5 + 12 + 13)
= ^X30
= 15
s - a = 15 — 5 = 10
s_[, = 15_12 = 3
5 - c = 15 - 13 = 2
SUPPLEMENTARY TOPICS
201
s(s-a) {s-b)(s -c)= 15.10.3.2
,Sa = Vs {s -a){S'- h) (s -c) = V15.IO.3.2
= V9OO
= 30 sq. in.
3. Instead of multiplying all these numbers together, it
is easier to factor them, thus:
V15. 10-3.2 = V3.5.2.5.3.2
= V32.52.22
= 352
= 30
4. Another illustration.
Given a = 15"
b = 18"
c = 21"
Find S.
.s =
S = Vs(s -a)(s-
i (a + b-{-c)
-1- (15 + 18 + 21)
i (r>4)
27
s
s
s
hKs - c)
a = 27 - 15
6 = 27 - 18
c =27-21
12
9
6
S (.s - a)(.s - b){s - c) = 27.12.9.6
^Sa = Vs(s-a){8-b)is-c) = V27.12.9.G
= V3.3.3 X3.2.2 x3.3 x2.3
= V32. 32. 22. 32-2. 3
= 3.32.3^2^
= 54\/6
= 54 X 2.449 +
= 132.24 + sq. in.
or
202
MODERN JUNIOR MATHEMATICS
. V27 X 12 X 9 X 6 = V17496
Sa = 132.2 +
17496.00 1 132.2 +
1_
23 I 74
69
262 I 596
524
2642 I 7200
5284
1916
5. Find the area of a triangle whose sides are 5, 6, and
7 inches respectively.
6. Find Sa if a = 9, 6 = 10,
and c = 11 ft.
7. An irregular field has its
successive sides 130, 50, 100,
and 80 rods. Its shorter diag-
^ onal is 120 rods. What is the
^•^^^^' area of the field?
C. LESS COMMON SOLIDS
There are three other soUds which we shall examine
briefly because, although important, they are less common
than others. They are the pyramid, cone, and sphere.
Only the right pyramid and right circular cone will be
considered.
L The Pyramid
1. How many bases has a pyramid?
2. What shapes may the base have?
3. What is the shape of the lateral faces?
4. The point at which the lateral faces meet is the vertex
of the pyramid.
SUPPLEMENTARY TOPICS
203
5. The altitude of one of the triangular faces is called
the slant height of the pyramid (Z).
Triangular
Pyramid
Rectangular
Pyramid
Hexagonal
Pyramid
6. A perpendicular Une from the vertex to the center of
the base is the height of the pyramid or its altitude,
7. If possible find a hollow prism and a hollow pyramid
with equal bases and equal height. Use the pyramid as a
measuring cup to fill the prism with water.
You will find that the pyramid must be filled three times.
For the prism, V^t = Bh
for the pyramid, Fpyr - \ Bh
To show the difference between V for prism and V for
pyramid they may be written Fpr and Fpyr respectively.
8. A rectangular pyramid has a base 8" square and a
height of 12". Find its volume.
9. The base of a pyramid is an equilateral triangle whose
edge is 4". The slant height is 6". Find the lateral area.
10. The area of the base of a pyramid is 20 sq. in.; the
altitude is 15 in. Find its volume.
11. The base of a pyramid is a regular hexagon 8 ft. on
a side; the height is 12 ft. Find its lateral area.
204 MODERN JUNIOR MATHEMATICS
12. What is the shape of each face of a pyramid? How
can you find the area of one face from the slant height and
the edge of the base?
13. The base of a regular hexagonal pyramid is 6" on a
side. The slant height is 10". Find the lateral area.
14. The base of a pyramid contains 75 sq. in. Its height
is 12 in. What is its volume?
n. The Cone
1. Draw and cut out a circle.
2. Along one radius cut from the circumference to the
center.
3. Lap over the two edges, at first just a little, then
one-fourth to one-half and until the two edges meet.
4. The resulting figures are cones.
6. You can readily see that the lateral surface of a cone
is a part of a circle.
If the radius of the base is r [ind the slant height of the
cone is I, then the
Lat. Sco = TT rl.
6. Just as the pyramid is J of the prism, so the cone is
-3 of the cylinder having the same base and height.
Since Fcyi = tt r%
T^co = 3 ^ r^h.
SUPPLEMENTARY TOPICS 205
7. A round tower 21 feet high is 10 feet in diameter and
is capped by a cone 12 feet high.
(a) The slant height
is the hypot-
enuse of a
right A. Find
its length.
(b)How many
square feet of
tin are re-
quired to
cover the
roof?
(c) What will it cost
to paint the
tower at $2.25 ;
per square,
and to paint
the roof at S2.00 per square?
8. On a bam floor is a pile of wheat. Naturally its
shape is a low cone. It is 8' in diameter and 2\' high.
How many bushels of wheat are in the pile if Ij cu. ft.
make 1 bushel?
9. The slant height of a cone is 4J ft. The radius of the
base is 2 ft. Fin'd the lateral area.
10. Find the total area of a cone whose slant height is 3"
and whose base has a radius of 2",
11. A cone 6 in. high has a base with a 4-in. radius. What
is its volume?
m. The Sphere
Sphere is the mathematical name of the toy of your
earliest childhood. If a sphere is cut in two equal parts,
two hemispheres (half-6pheres) are formed.
206 MODERN JUNIOR MATHEMATICS
1. The surface of a sphere is four times as large as a
circle with the same diameter.
Ssph = 47rr2
2. The volume of a sphere is the cube of the radius mul-
tiplied by i TT.
3. A steel ball is 10" in diameter.
(a) How many cu. in. are in its volume?
(b) Find its weight if the density of steel is 28 lb. per
cu. in.
4. In round numbers the diameter of the earth is 8000
miles.
(a) In round numbers find the total surface of the
earth.
(b) What is the ratio of the land area to the water
area?
(c) How many square miles of land are there?
(d) In round numbers find how many cubic miles
the earth contains.
6. Find the surfaces of the following spheres:
(a) r = 3|" (c) r =• 5 ft. 3 in.
(b) r = 4V' (d) r = Sf"
6. Find the volume of each sphere given in example 5.
IV. Patterns for Solids
1. What is a pyramid?
How does a rectangular pyramid differ from a triangu-
lar pyramid?
2. Polyhedron is a name given to any solid boimded by
SUPPLEMENTARY TOPICS
207
plane faces. This name comes from two Greek words, poly,
which means many, and hedron, which means base or face.
TpTRAHEDRON
Name all the polyhedrons that you know.
The faces of a regular polyhedron are regular polygons.
3. The regular polyhedron having four equilateral tri-
angles as faces is sometimes called a tetrahedron.
Since tetra is the Greek word which means four, why is
tetrahedron a good name for a triangular pyramid?
4. To construct these figures, draw the patterns on stiff
paper, cut out, and fold on the dotted lines. Paste together
with the flaps.
5. What kind of soUd has square faces?
Why may a cube be named a hexahedron?
Octahedron
6. A soUd having eight triangular faces is called an
octahedron.
208 MODERN JUNIOR MATHEMATICS
Which part of the name tells the number of faces?
Does this part mean the same number in octagon and
octave?
7. There are only two other regular polyhedrons. One
has twelve regular pentagons for faces; the other has
twenty equilateral triangles.
MISCELLANEOUS PROBLEMS
1. A girl wishes to make a shirt waist box out of a canned
goods box from a grocery. It measures 26" x 14" x 13".
How much Japanese matting is needed to cover the outside
of the box?
How much cretonne 27 in. wide is needed for Uning the
inside?
2. (a) How much paint is needed to paint four square
columns 3 ft. wide and 15 ft. high? (See ex-
ample 13, page 18.)
(b) The painter receives $.65 per hour and takes 2
hours for a square. How much does he receive?
3. The walls and ceiling of a room are tinted. How
much surface is covered if the room is 14' x 14' x 9'?
Allow for two doors 7' x 3J' and one window 7' x 4'.
4. How much tin is needed for a 5-lb. candy box which
is lOj" X 6f " X 3i"? Allow i inch for the overlappmg of
the lid.
6. A boy has made a tool chest 28" X 15" x 12". An-
other boy agrees to stain the sides and both sides of the lid
for him at 2^ per square foot. How much does the second
boy earn?
6. How many square feet of surface are in the walls and
ceiling of your living room at home? In the floor?
7. A Gold Dust box is 4" long, H" wide, and 6i" high.
How many square inches are in the paper label pasted all
over it?
8. A box of pepper is 3|" high. The bottom is IJ" x 1".
(a) How much paper is used for the label pasted all
around the sides?
(b) How much tin is used in the box?
9. A water tank is 3| ft. long, 2f ft. wide, and 2 ft.
deep. How many square feet of zinc are required to line
209
210
MODERN JUNIOR MATHEMATICS
the four sides and base, allowing 2 sq. ft. for overlapping
and for turning the top edge?
10. (a) Just inside a rectangular garden 40' X 32' is a
walk 4' wide. How many square yards of
cement pavement are in the walk?
(b) How many square feet of the garden are left for
flowers?
(c) What part of the entire garden is used for flowers?
11. A pasture 120 ft. long and 100 ft. wide has an 8-ft.
gate in one end. How much wire will be needed to fence it?
12. A picture 15" by 12" is to be framed with 1" molding.
How many feet of molding will it take?
13. Material 50 inches wide was bought for draperies.
It is discovered after the curtains are made that a piece
18 inches long is left, enough to make half of a valence
18" X 100". All the 50-inch material had been sold, but
the same pattern in 30-inch material can be bought. How
much of the 30-inch drapery is needed to finish one valence
and to make another?
14. Find the square roots of the following:
(a)
5476
(b)
233289
(c) 5776
(d)
1849
(e)
532900
(f) 5041
(g)
374544
(h)
7225
(i) 9025
(J)
5625
(k)
7569
(1) 331776
(m)
5329
(n)
6084
(o) 356409
(P)
8649
(q)
736164
(r) 793881
(s)
4562496
(t)
18190225
(u) 39204
16. (a) What is a pendulum? How is it used?
(b) A pendulum swings from
vD some fixed point of sup-
port as 0, through the arc
of a circle as AA\ The
time it takes for the
pendulum to swing
MISCELLANEOUS PROBLEMS 211
through this arc depends upon the length of
the pendulum or OA,
(c) A formula has been found by which one can find
the time it takes any pendulum to swing
through its arc. _
The formula is, ^ = tt A/ --
V g
t is the time or number of seconds,
TT is 3.1416 as used in a circle,
I is the number of feet in the length of the pen-
dulum,
g is the force of the attraction of the earth or
gravity, g has an approximate value of 32.
(d) Read this formula in EngUsh.
(e) How many seconds does it take a pendulum 4 ft.
long to swing through its arc?
(f) Find the time for a 3-ft. pendulum; a 2-ft. pendu-
lum; a 1-ft. pendulum.
16. (a) If a stone is dropped from a second story win-
dow 16 ft. above the ground and another is
dropped from a window four times as high, 64
feet above the ground, do you think it takes
four times as long for the second stone to reach
the groimd? That is the natural conclusion,
but it is not true.
(b) There is a formula by which the time it takes a
body to fall may be found.
The formula is, ^ = i /— .
V g
t is the time in seconds,
s is the number of feet in the distance through
which the body must fall,
g is the force of gravity, which is 32.
(c) What does 2 s mean?
212 MODERN JUNIOR MATHEMATICS
(d) Read the formula in English.
(e) How many seconds does it take a stone to fall
16 feet?
(f) Find the time it takes for one to fall 64 feet.
(g) Compare the two times.
(h) How long does it take a body to fall 8 feet?
(i) In a storm a ball was loosened on the top of a
church spire 320 feet high and fell to the groimd.
How many seconds was it in falling?
17. The vertex angle of an isosceles triangle is 36°, what
is the size of each base angle?
18. In a triangle ABC, angle B is twice angle A, and
angle C is three times angle A, How many degrees are in
each angle?
19. One acute angle of a right triangle is 28° 40'. What
is the size of the other acute angle?
20. One angle of a triangle contains 50° 30'; another
contains 88° 40'. What is the size of the third angle?
21. In a triangle ABC, Z A = 48° 50'; Z J? = 65° 30'.
^Vhat is the size of the exterior angle at vertex C?
22. A wire is fastened at one end to a telegraph pole,
18 ft. from the ground, and at the other to a stake at the
level of the ground, 14 ft. 6 in. from the foot of the pole.
How long is the wire?
23. A May pole, 10 ft. high, is set in a circle whose radius
is 8 ft. How long must be the streamers fastened at the
top of the pole in order that they may reach the edge of
the circle?
24. If the foot of a ladder 24 ft. long is 12 ft. from a house,
how far up the side of the house does
the ladder reach?
26. In a right triangle, a = 67.2'
and c = 110'. Fmd 6.
- 26. From the data given in the
figure, find a.
MISCELLANEOUS PROBLEMS 213
27. The sides of a triangle are 37.5 ft., 90 ft., and 97.5 ft.
Classify the triangle as to its sides and angles.
28. (a) A regular hexagon is inscribed in a circle whose
radius is 12 ft. Find the radius of the circle
inscribed in the hexagon.
(b) What is the perimeter of the hexagon?
29. (a) A girl crochets a medallion in the shape of an
isosceles triangle. It is to be 8 inches long at
the base and 3 inches high. How long must
each edge be?
(b) She wants two others, each 5 inches at the base
and 1| inches high. How long is each side of
the smaller ones?
30. A room is 18' X 15'. What is the shortest distance
from one corner to the diagonally opposite one?
31. What is the largest square rod that can be cut out of
2|" round stock?
32. (a) Velvet, one yard wide, is cut on the bias. It is
sold by the measure on the straight edge.
(b) How many inches on the straight must be pur-
chased in order to have a bias band 8 inches
wide?
(c) How long will the band Xi^'l
214 MODERN JUNIOR MATHEMATICS
33. A balloon is 1500 ft. in the air. If a stone is dropped
from it, how long will it take to reach the earth? Use formula
t
Vt
34. (a) From the data given in the figure find the height
of the tent pole, a.
(b) How many square yards
of canvas are needed
to make the tent?
36. (a) A girl has a flower bed
■jy^ in the shape of an
equilateral triangle
8 ft. on a side. How many tulip bulbs must she
buy if she allows 36 square inches for each tulip?
(b) What will they cost at 5 cents each?
36. City streets intersect in such a way that there is a
triangular park formed 90 ft. on a side. How many square
feet of sod are needed for it?
37. (a) In these circles hexagons and triangles are in-
scribed and triangles are circumscribed about
them in two ways. Draw the figures and letter
all the points of intersection.
(b) Find all the equal lines.
(c) Find all the equal triangles.
(d) Can you find any equal rhombi?
(e) How many degrees are in Z ABC? Z FBDf
Z ABFf Z CBD?
(f) Without using the protractor find the number of
degrees in each angle.
(g) Find all the equal angles.
MISCELLANEOUS PROBLEMS
38. A ladder 38 ft. long is resting against a wall. If the
foot of the ladder is 7 ft. 3 in. from the wall, how far
above ground does the ladder reach?
39. A steel bridge has a truss ABCD.
AS = 20 ft. ED ^6 ft. 4 in.
DC = 12 ft.
Find the length of AD and DF.
40. (a) The following are dimensions and areas of rec-
tangles. _ Find the ones missing and the (^wk^.-
eters.
216
MODERN JUNIOR MATHEMATICS
(1
(2
(3
(4
(5
(6
(7
(8
(9
(10
(11
(12
(13
(14
(15
(16
(17
(18
(19
(20
Area
6 xV + 14 ary + 8
327r2 + 887ri2 + 20i?»
135 x^ + 51 xy 4- 2 2/«
12 a* + 46 a6 + 42 6^
2 a* + 12 a + 18
28 ^ + 80 <w + 48 w2
35 X* + 34 0:2/ + 8 2/2
36 a* + 69 a6 4- 30 ¥
a«+462+9c2 4-4a6 + 6ac+ 126c
\x^ + \xy-\-\y^
Length
2a6 + 5
7a + 26
3x2/ + 4
87r + 2i2
45 X + 2 2/
27ri2 + 6
8x + 92^
2a + 6
3a + 56 + 6c
81 + 3 A;
2X+32/ + 1
3a 4-6 + 5 c
3a4-5x4- 6
a4-264-3c
ia4-i6
Width
a6 4-3
3a4-46
27ri2 + 2
3a4-76
7x4-32/
4e4-8w
5x4-22/
4a4-56
2a4-2 6 4-c
61 4- 2 A;
0? 4-22/ 4-3
2a4-64-3c
2a4-x4-7
ia4-i6
\x-\-\y
Perimeter
(b) Which of these rectangles are squares?
41. A post 7 ft. high casts a shadow 4 ft. long at the
same time of day that a tree casts one 32 ft. long. Find
the height of the tree.
42. A boy 5 ft. tall makes a shadow 10 ft. long. At the
same moment the shadow of a building is 125 ft. How
high is the building?
43. By the shadow method, measure the heights of trees
or buildings in the vicinity of your school.
44. A and D are right angles.
If AB = 33 ft.
AC = 15 it.
MISCELLANEOUS PROBLEMS
217
and CD = 125 ft.
what is DEf the width of the river?
45. (a) Find the dimensions and perimeters of the follow-
ing rectangles. Check results.
(1
(2
(3
(4
(5
(6
(7
(8
(9
(10
(11
(12
(13
(14
(15
(16
(17
(18
(19
(20
Area
idx + i ay + laz
i a(rn + n) + i b(m + n)
(x + 1) (x + 3) 4- (x + 2) (x + 3)
(a + 6) (a + c) + (a + d) (a + c)
(a + 2)' + 5(a + 2)
ia^ + iab + ib'
25 X* + 65 aaj + 42 a*
10» + 6 (10) 4- 9
196 + 28 X + X*
25 a« + 60 a2/ + 36 y^
63 x* + 58 xy + 7 y*
92 + 9 (9) + 20
a* + f a + i
289 a* 4- 136 a + 15
62 + 12 (6) + 35
x« + f X + i
a? + 5(J) + 6
81 x* + 72 ox + 16 a«
4a2 + 96* + c2 + 12a6 + 4ac + 6fec
Length
Width
Perimeter
(b) Which of these rectau^ei^ ^xfe %q^^\^.
218 MODERN JUNIOR MATHEMATICS
46. (a) A boy wishes to measure the height of the school
building. His only instruments are a tape line
and a 15-in. ruler. He holds the ruler ver-
tically in front of him, walks to such a posi-
tion that he can just sight top and bottom of the
building over the top and bottom of the ruler.
A classmate measures for him his distance from
the building and finds it is about 79.9 ft. The
boy^s eye is 5 ft. from the groimd. The bottom
of the ruler is 24 in. from his eye.
^ (b) Given AB = 5 ft.
BC = 79.9 ft.
To find AC to the nearest
integer.
(c) What kind of triangles are
AEF and ACDf
Given AE = 24 in.
EF = 15 in.
Find CD J the height of the building.
47. On a map Reims is on a line 7f " east of Paris and 2f "
north of it. The map is drawn on a scale of 1" to 10 mi.
What is the distance from Paris to Reims?
48. On a blue print drawn to a scale of |" to a foot, what
lengths are represented by the following lines?
(a) If" (d) A" (g) 121- (j) 3i"
(b) 5f " (e) 7i- (h) 15f" (k) lA"
(c) 2A" (f) lOi" (i) 41" (1) 8A"
49. (a) If it takes 2 cu. yds. of gravel for a 30-ft. sidewalk,
how much will it take for a 42-ft. walk?
(b) A sack of cement lays 1 ft. of sidewalk. If it is
spread thin and 8 sacks are used for 9 ft. of
walk, how much will it take for 18 ft.? For
20 ft.? For 30 it.? ¥oT\*liU
MISCELLANEOUS PROBLEMS
219
60, (a) A garage is 20' by 18' by 10'. How many gquarea
are in its lateral area?
(b) The roof is in the shape of two triangles at the
ends and two trapezoids at the sides of the
If it extends 2 ft. over the sides of
the building, what is the length of the base of
the triangular parts of the roof? Of the trape-
zoidal parts?
(c) The sloping edge {hip rafter) is 16 ft. long. The
ridge pole is 2 ft. long. What is the altitude of
each triangular section? of each trapezoidal
section?
(d) How many square feet of roofing are needed?
61. A baking powder box is 5" high. Its diameter is 3".
What is the size of the label?
Allow §" for overlapping of the lid and find how much
tin the box takes.
62. A piece of stove pipe is worn out. It is 23" long and
4§" in diameter. How much sheet iron is needed for a new
piece?
53. A cylindrical salt box is 5|" high and 3|" in diam-
eter. How lai^ must the label be to surround the box?
How much cardboard is needed for the whole bo^l
220 MODERN JUNIOR MATHEMATICS
64. Measure some object near your school which is cylin-
drical in shape. Calculate the lateral and total areas.
65. Make the same measurements and calculation for
some cylindrical object in your home or place where you
work.
56. What is the area of a watch crystal if its diameter
is If"?
57. A round rattan work basket has a radius of 3^".
What is the area of the top of the hd?
What is the total area of the lid, if it extends 1^" over the
basket?
68. A preserving kettle is 13" in diameter. How much
tin is needed for a lid to cover it?
69. (a) A pint cup is 2| inches high and 11 inches around.
How much tin is needed for it?
(b) Show that it holds a full pint.
60. A measuring cup is 6 cm. high and 22f cm. around.
How much aliuninimi is needed for the cup?
REVIEW QUESTIONS
On Chapter One
1. Define the following terms: parallel, perimeter,
formula, coefficient.
2. Describe a cube; an oblong block.
3. Give the formula for the perimeter of a square; of a
rectangle.
4. Of what use is the process of substituting nimierical
values in a formula?
On Chapter Two
1. Define the following terms: angle, right angle, straight
angle, perpendicular, acute angle, obtuse angle, adjacent
sides, surface, area, lateral area, the square of a number,
the square root of a nimiber, factor, prime factor, exponent.
2. Inquire from a carpenter and draftsman the uses of a
T square.
3. Describe the following: square, rectangle, parallelo-
gram, rhombus, rhomboid, radical sign.
4. Give the formula for the area of a square; of a
rectangle.
6. How does a decimal fraction differ from other frac-
tions?
6. Is the square of a number greater or less than its
square root?
On Chapter Three
1. Define the following terms: diagonal, vertex, comple-
mentary angles, interior angles of a triangle, exterior angles
of a triangle, hypotenuse, leg of a right triangle, legs of an
isosceles triangle.
221
222 MODERN JUNIOR MATHEMATICS
2. What is the sum of the interior angles of a triangle?
3. Classify triangles according to the size of their angles.
Describe each kind.
4. Classify triangles according to the relative lengths of
the sides. Describe each kind.
6. Give the formula for the area of a triangle.
6. Two triangles have two sides of one equal respectively
to two sides of the other. In the first triangle these sides
form an angle of 90°, and in the second an angle of 60°.
Which triangle is the larger? Why?
On Chapter Four
1. Define the following terms: circle, diameter, semi-
circle, circumference, arc, intersect, base angles and vertex
angle of an isosceles triangle, bisect, bisector.
2. Describe an angle inscribed in a circle; one inscribed
in a semicircle.
3. A square and an octagon are inscribed in a circle.
Which has the longer perimeter? Why?
4. Why are the maps on ordinary railroad time tables
distorted? Are the real railroads in as straight lines as the
maps represent them?
On Chapter Five
1. Define the following term: circumscribed circle.
2. Describe all the special features of an isosceles tri-
angle and its altitude.
3. Who was Pythagoras? What is the Pythagorean
theorem?
On Chapter Six
1. Define the following terms: parallel lines, transversal,
interior angles of parallel Unes, exterior angles of parallel
lines, alternate interior angles, corresponding angles, vertical
angles, suppJementary angles.
REVIEW QUESTIONS 223
2. When two parallel lines are cut by a transversal,
which angles are equal and which are supplementary?
3. If the vertex of a triangle is folded over to the foot
of a perpendicular drawn from that vertex, what is the rela-
tion of the fold to the base of the triangle?
On Chapter Seven
1. Define the following terms: median, midjoin.
2. In a parallelogram which Unes and which angles are
equal?
3. Draw the two diagonals and answer the above ques-
tion.
4. How do the answers to these questions change when
the parallelogram is a rhombus? a rectangle? a square?
On Chapter Eight
1. Define the following terms: ratio, equation, graph,
monomial, binomial, trinomial, polynomial.
2. Explain percentage in terms of ratio.
On Chapter Nine
1. How does the sum of the interior angles of a rhom-
boid compare with the sum of those of a rectangle?
2. What is the formula for the diagonal of a square in
terms of its side?
3. Of what general formula is that for the diagonal a
special case?
On Chapter Ten
1. Define the following terms: similar figures, propor-
tion.
2. What are some of the practical uses of similar figures?
3. Under what conditions are two trijangles similar?
4. Give all the interesting facts you can about Thales.
6. Which man do you think did the most for mathe-
matics, Thales or Pythagoras? Why?
224 MODERN JUNIOR MATHEMATICS
6. Describe the following instruments and their uses:
transit, sextant, quadrant, baculus.
On Chapter Eleven
1. Describe the following figures: trapezoid, isosceles
trapezoid.
2. Give the formula for the area of a trapezoid.
3. How is this formula related to that for the area of a
triangle?
4. Examine as many geometrical designs as possible in
linoleum, wall paper, etc., and see if certain polygons are
used much more than others. List the polygons in order
of the frequency of their use.
On Chapter Twelve
1. Define the following terms: concentric circles, cylin-
der, cylindrical surface, plane surface.
2. What is the test for a plane surface?
3. What is the ratio of the diameter to the circumference
of a circle? Give a brief history of this ratio.
4. What is the formula for the circumference of a circle?
For the area of a circle?
On Chapter Thirteen
1. Describe a prism.
2. Describe triangular, rectangular, and hexagonal
prisms.
3. What is the difference between a cylinder and a
prism?
SYMBOLS AND ABBREVIATIONS
II = parallel, is parallel to.
P = perimeter.
B = area of base.
6 = base (base line).
Lat. = lateral.
e = edge.
I = length.
w = width.
h = height.
D, HI = square, squares.
□,[!] = rectangle, rectangles.
^ly HJ = parallelogram, parallelograms.
O, ZI7 = rhombus, rhombi.
A, A = triangle, triangles.
Z , ^ = angle, angles.
O = trapezoid,
rt. = right.
S = surface or area of surface.
v^ = square root of.
comp. = complementary,
sup. = supplementary,
hyp. = hypotenuse,
alt. = altitude.
O, © = circle, circles,
isos. = isosceles.
r^, r^ = arc, arcs.
.*. = therefore.
± , -b = perpendicular, or is perpendicular to, perpendiculars.
a,b,c^ sides of scalene triangle.
a, 6 = legs of right triangle.
^ 225
226 MODERN JUNIOR MATHEMATICS
c == hypotenuse of right triangle.
( ) = parentheses.
-- = sunilar, is similar to.
7r= pi, ratio of circumference to diameter of circle.
V = volume.
c = circumference.
d = diameter.
r = radius.
cu. = cube.
CO. = cone.
cyl. = cylinder.
pr. = prism.
pyr. = pyramid.
sph. = sphere.
APPENDIX
A. MATHEMATICS CLUBS
Mathematics clubs among students in colleges and high
schools have proved their worth. Nearly all of the pro-
gressive schools of these grades number such organizations
among their student activities.
It was the author's privilege to organize and direct a
mathematics club among the boys of grades ten to twelve
in the high school in which she formerly taught mathe-
matics. Three years ago, she suggested, as an experiment,
the organization of a similar club in one of the junior high
schools of Columbus. Under the direction of a mathe-
matics teacher, the first club of ninth grade pupils has be-
come the Alpha Chapter of the original EucUdean Club,
for the Beta and Gamma Chapters have been organized in
the eighth and seventh grades respectively.
The purpose of a mathematics club is to promote interest
in the study of mathematics, to give the pupils glimpses of
the future, which serve as incentives to continue the study,
and to furnish an outlet for their social instincts.
Stories from the history of mathematics, magic squares
and circles, mathematical fallacies, and other recreations
furnish interesting material for club programs.
The topics may be the same for clubs in the several grades
but the treatment will be different in each. To show this
difference a program for each grade in the same topic is
given:
Topic — Magic Squares.
I. Ninth Grade.
1. History of Magic Squares.
227
228 APPENDIX
2. How to Make Magic Squares (with an odd number
of sides).
3. How to Make Magic Circles.
II. Eighth Grade.
1. How to Make a Magic Square (with odd number of
sides).
2. How to Make a Magic Circle.
3. Some Interesting Facts about Magic Squares.
III. Seventh Grade.
1. A Magic Square, 3 numbers on a side.
2. A Magic Square, 5 numbers on a side.
3. A Magic Square, 7 mmibers on a side.
4. Board Work with Magic Squares.
These sample programs for the ninth grade may be sug-
gestive:
I. 1. Euclid.
2. Some Interesting Things about a Billion. (White.)
3. How to Write 100 in Several Ways. (T. C. Record,
November, 1912.)
4. Some Questions. (Original by pupil.)
II. 1. Familiar Trick with Dice. (White.)
2. Mathematical Advice to a Building Conunittee.
(White.)
3. Puzzle of the Camels. (White.)
4. Pythagoras.
III. 1. MultipUcation on Fingers.
2. Russian MultipUcation (only table of 2's need be
known). (School Science and Mathematics,
April, 1919.)
3. Ship Carpenter's Puzzle. (White. — Presented one
meeting. Solution given the next.)
4. Story of Flatland (told by a pupil^ .
APPENDIX 229
Topics taken from the following subjects will be inter-
esting for other programs:
1. Napier's Rods.
2. Descartes' Life.
3. A Fairy Tale. (School Science and Mathematics.)
4. A Nimiber Trick. (White.)
5. A Riddle. (Jones.)
6. To Prove 1-2.
7. Poem: A Yomig Lady and Her Lover. (Jones.)
8. Some Interesting Questions. (Jones.)
9. Remarkable Numbers. (Teachers' College Record,
November, 1912, or Jones.)
10. Trisecting an Angle.
11. Duphcating the Cube.
12. Squaring the Circle.
13. Fourth Dimension.
14. Mathematical Symbolism.
15. The Golden Section. (May, 1918, of American
Mathematical Monthly.)
16. Proofs of Pythagorean Theorem. (Monograph —
D. C. Heath & Co.)
17. Use of Mathematics in Science
18. History of Arithmetic.
19. History of Algebra.
20. What is a Straight Lme?
21. Computing Machines.
22. History of Pi.
23. The Algebra of Al-Khowarizmi.
24. Hindu-Arabic Numerals.
25. Paper Folding.
26. Fallacies of Arithmetic.
27. Opportunities Open to Students of Mathematics.
28. Women Mathematicians. (March, 1918, of
A. M. M.)
29. Game of "Nim." (JAatdi, \^\^, ^1 k.^^.^i^^»
230 APPENDIX
30. Chinese Rings. (March, 1918, of A. M. M.)
31. A, B, andC. The Human Element in Mathematics.
(S. Leacock^s ''Literary Lapses.'')
32. Logarithms.
33. The Oldest Mathematical Work.
34. Great Mathematicians, as:
(a) Newton, Astronomer and Mathematician.
(b) Archimedes, Inventor and Mathematician.
Pupils become so interested in the mathematics clubs
that they bring in material, invent games and tricks, and
even write and dramatize mathematical plays.
The following books and magazines are helpful:
1. Abbott. Flatland. Little, Brown, $0.60.
2. Andkews. Magic Squares and Cubes. Open Court
Pub. Co., $1.50.
3. Ball. History of Mathematics. Macmillan, $3.25.
4. Ball. Primer of the History of Mathematics. Mac-
millan, $0.60.
5. Ball. Mathematical Recreations. Macmillan, $2.25.
6. Jones. Mathematical Wrinkles. S. P. Jones of Gun-
ter, Texas, $1.65.
7. Smith and Karpinski. The Hindu-Arabic Numerals.
Ginn, $1.40.
8. White. Scrap-book of Elementary Mathematics. Open
Court Pub. Co., $1.00.
9. School Science and Mathematics.
Smith and Turton. (Magazine and membership in
Central Association of Science and Mathematics
Teachers, $2.50 per year. Chicago.)
10. Teachers' College Record — Columbia University, $1.50
per year.
11. American Mathematical Monthly ^ Journal of Mathe-
matical Association of America. (Magazine and
membership, $3.00 per year. Chicago.)
12. Smith. Number Stories oiLoii^k^o. CjcvMi&.^Q. %^A^,
APPENDIX 231
B. TABLES
Linear Measure for Length
12 inches (in.) = 1 foot (ft.)
3 feet = 1 yard (yd.)
5J yards or 16J feet = 1 rod (rd.)
320 rods = 1 mile (mi.)
1 mi. = 320 rd. = 1760 yd. = 5280 ft. = 63,360 in.
A hand = 4 in., used in measuring the height of horses.
A fathom = 6 ft., used in measuring the depth of large
bodies of water.
A knot = 1.15 mi., used in measuring distances at sea.
Square Measure fpr Surface
144 square inches (sq. in.) = 1 square foot (sq. ft.)
9 square feet = 1 square yard (sq. yd.)
30J square yards = 1 square rod (sq. rd.)
160 square rods = 1 acre (A.)
640 acres = 1 square mile (sq. mi.)
1 A. = 160 sq. rd. = 4840 sq. yd. = 43,560 sq. ft.
A section = 1 square mile
A square = 100 square feet, used in roofing, flooring, and
painting.
Cubic Measure for Volume
1728 cubic inches (cu. in.) = 1 cubic foot (cu. ft.)
27 cubic feet = 1 cubic yard (cu. yd.)
A cord = 128 cu. ft., used in measuring wood. It is
usually a pile 8 ft. by 4 ft. by 4 ft.
Measure of Capacity
A. Liquid Measure
4 gills (gi.) = 1 pint (pt.)
2 pints = 1 quart (qt.)
4 quarts = 1 ^aWoiv V??J^^
232 APPENDIX
1 gal. = 231 cu. in.
1 barrel (bbl.) = 31§ gal.
1 hogshead = 63 gal.
In commerce barrels and hogsheads vary in size.
1 gallon of water weighs about 8^ pounds.
1 cubic foot of water weighs about 62§ pounds.
B. Dry Measure for Fruits, VegetableSj and Grain
2 pints = 1 quart (qt.)
8 quarts = 1 peck (pk.)
4 pecks = 1 bushel (bu.)
1 bushel = 2150.42 cu. in. or about 1| cu. ft.
Avoirdupois Weight
16 ounces (oz.) = 1 pound (lb.)
2000 pounds = 1 ton (T.)
1 hundredweight (cwt.) = 100 pounds.
1 pound avoirdupois = 7000 grains.
1 long ton = 2240 pounds, used in weighing coal and ores
at the mines.
Standard Weights
1 bu. of wheat . . . . = 60 lb. 1 bu. of white po-
1 bu. of com, in the tatoes = 60 lb.
ear = 70 lb. 1 bu. of sweet po-
1 bu. of corn, tatoes :.. = 551b.
shelled = 56 lb. 1 bu. of com meal. = 48 lb.
1 bu. of oats = 32 lb. 1 bu. of clover seed = 60 lb.
1 bu. of rye = 56 lb. 1 bbl. of flour =196 lb.
1 bu. of barley. . . . = 48 lb. 1 bbl. of pork = 200 lb.
1 keg of nails =\^\fc».
APPENDIX 233
Troy Weights for Jewels and Precious Metals
24 grains (gr.) = 1 pennyweight (pwt.)
20 pennyweight = 1 ounce (oz.)
12 ounces = 1 pound (lb.)
1 pound troy = 5760 grains
Time
60 seconds (sec.) = 1 minute (min.)
60 minutes = 1 hour (hr.)
24 hours = 1 day (da.)
7 days = 1 week (wk.)
365 days = 1 year (yr.)
A leap year = 366 days.
A business year usually is 360 days or 12 months of 30
days each.
A decade = 10 years.
A century = 100 years.
Counting
12 things = 1 dozen (do'z.)
12 dozen = 1 gross
A score = 200 things
Paper
24 sheets = 1 quire
20 quires = 1 ream
Paper is usually sold by the 1000 (M), by 500 (D), or by
the 100 (C) sheets.
In practice 1 ream == 500 sheets
1 quire = 25 envelopes or cards
Arcs of a Circle
60 seconds (") = 1 minute (')
60 minutes = 1 degree (°)
360 degrees = 1 cvicwroi^x^w^^.
234 APPENDIX
Angles
60 seconds (") = 1 minute (')
60 minutes = 1 degree (°)
90 degrees = 1 right angle (rt. Z ) or 1 quadrant
180 degrees = 1 straight angle (st. Z )
360 degrees = 1 perigon
United States Money
10 mills = 1 cent (i)
10 cents = 1 dime
10 dimes = 1 dollar ($)
Money Equivalents
England — 1 pound (£) = $4.8665 in U. S. gold coin
France — 1 franc (fr.) = 19.3 cents
Metric System
Linear Measure for Length
10 millimeters (mm.) = 1 centimeter (cm.)
10 centimeters = 1 decimeter (dm.)
10 decimeters = 1 meter (m.)
1 meter = 39.37 inches
1 yard = .9144 meter
Square Measure for Surface
100 square millimeters = 1 square centimeter (sq. cm.)
(sq. mm.)
100 square centimeters = 1 square decimeter (sq. dm.)
100 square decimeters = 1 square meter (sq. m.)
1 square meter = 1.196 square yards
1 square yard = .8S& s>c\u"a.T^ tji^X^t
APPENDIX 235
Cubic Measure for Volume
1000 cubic millimeters = 1 cubic centimeter
(cu. mm.) (cu. cm. or c.c.)
1000 cubic centimeters = 1 cubic decimeter (cu. dm.)
1000 cubic decimeters = 1 cubic meter (cu. m.)
1 cubic meter = 1.308 cubic yards
1 cubic yard = .765 cubic meter
Measure of Capacity
1 liter (1.) == .908 dry quart
1 dry quart = 1.1012 liters
1 liter = 1.0567 liquid quarts
1 liquid quart = .94636 liter
Measure of Weight
1 gram (g.) = weight of 1 cu. cm. of water
= .0022 poimd
1 pound = 453.59 grams
C. A PROTRACTOR AND ITS USE
There are many kinds of protractors but the most satis-
factory kind is a combined ruler and protractor. It should
be made of celluloid because of its transparency. One edge
of the ruler should be marked in EngUsh imits and the
other in metric units. One end should be marked in tenths
of inches.
The Granville Combined Ruler and Protractor has one
end marked in fiftieths of an inch and its width is exactly
the square root of 2 inches.
The following figure shows how to dia.^ ^\s.^\ns^^ ^^:^ ,
236 APPENDIX
Given any line AB. Select some point, C, in the line and
place the center of the protractor at this point. Shde the
protractor around until its 60° line exactly coincides with
AB. Then draw a line along the edge CD. Z BCD is a
60° angle.
INDEX
PAGE
Acute angles 10
Acute triangles 37, 43
Addition. .102, 110, 112, 115, 116
Adjacent 7
Ahmes 188
Alternate-interior angles 73
Angles. . .8, 9, 10, 31, 32, 33, 34,
46, 51, 73, 80, 134
Angles of a triangle 32, 33
Arc 45
Archimedes 188
Area 13, 14, 38, 39, 83,
109, 110, 111, 113, 115, 116,
133, 185, 186, 187
Baculus 156
Bar graph 99
Base 3
Base angles 46
Bettinus 150, 155, 160
Binomial 105, 110
Bisection 51
Carpenter's square 10, 76
Changing common fractions to
decimal form 28
Checking 114, 116, 127, 133
Circle. ..43, 55, 56, 181, 185, 190
Circumference 45
Circumscribed 56
Coefficient 5, 14, 69, 112
114, 127
Complementary angles 34
Cone 204
Constructions . . 43, 47, 77, 79, 80,
170, 171, 172, 175
Corresponding angles of paral-
lel lines 73
Cube 2,7
Cylinder 179, 187, 190, 195
Cylindrical surface 180
Decimal fractions 25
Designs 56-59, 81, 170, 172
174, 177
Diagonal 31, 139
Diameter 43, 81
Dimensions 2, 119, 128, 129
Distance 50
Division 119, 121, 123, 124,
125, 128
Division sign 119
Dodecagon 168
Equation 63, 92
Equiangular 37
Equilateral 37, 45
Equipment xiii
Exponent 20, 122
Exterior angles of parallels ... 73
237
238
INDEX
Exterior angles of polygons ... 35
Factors 19, 112, 122, 124, 129,
130, 133
Form 1
Formulas 4, 5, 14, 15, 16, 36,
38, 41, 63, 103, 105, 106, 108,
119, 162, 182, 185, 186, 191,
195, 198, 200, 203, 204, 206
Fractions 25, 26, 27
Garfield 62
Graphs 66,96,99
Hero's formula 200
Hexagon 165, 167, 170
Hexahedron 207
Homologous parts 144
Hypotenuse 38, 136
Inscribed 55, 166
Interior angles of parallels 73
Interior angles of polygons ... 35
Intersection 45, 55
Irrational numbers 67
Isosceles 37, 46, 53
Lateral surfaces 3, 17
Lines 47,51, 53,80
Literal numbers 102
Mathematics clubs 227
Measurement 1, 3, 13
Mechanical method 24
Median 81
Midjoin 81
Monomial 105
MultipUcation 102-116, 127
Numbers 19
Oblongs 2, 7, 14
Obtuse angle 10, 11
Obtuse triangle 37, 45
Octagon 165, 171
Octahedron 207
Oughtred 120
Parallel 2, 6, 9, 71, 72, 75
Parallelogram 11, 60, 80, 83,
134, 143
Parallelopiped 16
Pell 120
Pentagon 165, 174, 176, 177
Percentage as ratio 98
Period 24
Perimeter ... 3, 4, 5, 101, 103, 106,
109, 110, 111, 113, 115, 116
Perpendicular. 9, 47, 48, 49, 50, 54
Pi 181,188
Polygon 165
Polyhedron 206
Polynomial 105
Prime factors 20, 21
Prime lettering 144
Prism 194
Protractor. .34, 39, 43, 48, 69, 235
Ptolemy 189
Pyramid 202
Pythagoras 62
Pythagorean 62, 66, 136,
137, 139
Quadrant 150
Quadrilateral 11, 79, 134, 165
Radical sign 21
Ratio ... 90, 91, 96, 134, 135, 142,
146, 148, 181, 192
Reading angles 31, 32
Reading lines 7
Rectangle 5, 7, 9, 12, 13, 14,
35, 79, 82, 87, 103, 107, 108,
118, 123, 125, 142
'Ref^xAdx 'joV^^oxL ,"\SRi,\^
INDEX
239
Review questions 221
Rhomboid 12
Rhombus 12,86
Right angle 8,9,10
Right triangle 37, 38, 43, 60,
61, 137
Rope stretchers 62
Scale drawing 94, 102
Scalene 37, 39, 47, 54, 77
Sextant 150
Similar figures 142
Sphere 205
Spherical surface 180
Square 5, 7, IJ, 16, 31, 80, 101,
117, 118, 139
Square root 19, 30, 64, 137,
138, 199, 201
Square root of 2, 3, and 5 . . . . 66
Squares of numbers 22-28
Straight angle 8
Subscripts 49
Substituting numerical values, 4,
5, 15, 16, 17, 40, 41, 64, 67,
89, 109, 113, 114, 121, 162,
163, 176, 186, 187, 197, 201
Subtraction 125, 126
Supplementary angles 73
Surface 15, 16
Symbols 3, 9, 12, 21, 31, 45,
109, 144, 225, 226
Tables 231
Term 104
Terminating decimals 28
Tetrahedron 207
Thales 44,49,151
Theorem 62
Times sign 5
Transit 150, 154
Transversal 72
Trapezoid 161
Triangle .... 31, 35, 37, 43-46, 53,
60, 136, 145, 146, 165, 200
Trinomial 105
T-Square 10,70
Unmeasured lines 101
Vertex 31, 32
Vertex angle 47
Vertical angles 73
Volume 194,195
To avoid fine, this bode should be returned o
or before the date last stamped below
-^
Cf^^
A.
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