# Full text of "Geometry (TI Activities), Teacher’s Edition"

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```CK-12 Foundation

CK-12 Texas Instruments
Geometry Teacher's Edition

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Printed: August 2, 2011

/lexboo<

next generation textbtwks

Editor

Lori Jordan

www.ckl2.org

Contents

1 TE Introduction to Geometry -

TI 1

1.1 Geometry TI Resources Flexbook 1

2 TE Basics of Geometry - TI 2

2.1 Midpoints in the Coordinate Plane 2

2.2 Vertical and Adjacent Angles 5

3 TE Reasoning and Proof - TI 8

3.1 Conditional Statements 8

4 TE Parallel and

Perpendiculuar Lines - TI 12

4.1 Parallel Lines cut by a Transversal 12

4.2 Transversals 13

4.3 Perpendicular Slopes 15

5 TE Congruent Triangles - TI 17

5.1 Interior and Exterior Angles of a Triangle 17

5.2 Congruent Triangles 20

5.3 Triangle Sides and Angles 24

6 TE Relationships within

Triangles - TI 27

6.1 Perpendicular Bisector 27

6.2 Hanging with the Incenter 29

6.3 Balancing Point 31

6.4 Hey Ortho! What's your Altitude? 33

7 TE Quadrilaterals - TI 37

7.1 Properties of Parallelograms 37

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7.2 Properties of Trapezoids and Isosceles Trapezoids 39

7.3 Properties of Rhombi, Kites and Trapezoids 40

8 TE Similarity - TI 43

8.1 Constructing Similar Triangles 43

8.2 Side-Splitter Theorem 46

8.3 Perspective Drawing 48

9 TE Right Triangle

Trigonometry - TI 52

9.1 The Pythagorean Theorem 52

9.2 Investigating Special Triangles 56

9.3 Ratios of Right Triangles 59

10 TE Circles - TI 63

10.1 Chords and Circles 63

10.2 Inscribed Angles Theorem 65

10.3 Circle Product Theorems 67

11 TE Perimeter and Area - TI 71

11.1 Diameter and Circumference of a Circle 71

11.2 From the Center of the Polygon 73

12 TE Surface Area and Volume -

TI 76

12.1 Surface Area of a Cylinder 76

13 TE Transformations - TI 79

13.1 Transformations with Lists 79

13.2 Reflections and Rotations 82

13.3 Perspective Drawing 84

111 www.ckl2.org

www.ckl2.org IV

Chapter 1

TE Introduction to Geometry
TI

1.1 Geometry TI Resources Flexbook

Teacher's Edition

Introduction

This flexbook contains Texas Instruments (TI) Resources for the TI-83, TI-83 Plus, TI-84, and TI-84 SE.
All the activities in this flexbook supplement the lessons in the student edition. Teachers may need to
download programs from www.timath.com that will implement or assist in the activities. Each activity
included is designed to help the teacher with each activity in the TI Resources Flexbook, Student Edition.
All activities are listed in the same order as the Student Edition.

There are also corresponding links in the 1st Edition of Geometry, 2nd Edition, and Basic Geometry.

• Geometry, first edition: http://www.ckl2.org/flexr/flexbook/805

• Geometry, second edition: http://www.ckl2.org/flexr/flexbook/3461

• Basic Geometry: http://www.ckl2.org/flexr/flexbook/3129

Any activity that requires a calculator file or application, go to http://www.education.ti.com/calculators/
downloads and type the name of the activity or program in the search box.

www.ckl2.org

Chapter 2

TE Basics of Geometry - TI

2.1 Midpoints in the Coordinate Plane

This activity is intended to supplement Geometry, Chapter 1, Lesson 4-

ID: 8614

Time required: 40 minutes

Topic: Points, Lines &: Planes

• Given the coordinates of the ends of a line segment, write the coordinates of its midpoint.

Activity Overview

In this activity, students will explore midpoints in the coordinate plane. Beginning with horizontal or
vertical segments, students will show the coordinates of the endpoints and make a conjecture about the
coordinates of the midpoint. This conclusion is extended to other segments in the coordinate plane.

Teacher Preparation

• This activity is designed to be used in a high school or middle school geometry classroom.

• The screenshots on pages 1-3 demonstrate expected student results.

• The Coordinate Midpoint formula for the midpoint of {x\,y\) and (x2,y2) is \ 2 2 , ^ 2 >2 )• This
can also be expressed as "The coordinates of the midpoint of a line segment are the averages of the
coordinates of the endpoints."

Classroom Management

• This activity is designed to be student- centered with the teacher acting as a facilitator while students
work cooperatively. The student worksheet provides a place for students to record their observations.

• Depending on student skill level, you may wish to use points with integer coordinates, or only positive
values.

• Note: The coordinates can display 0, 1, or 2 decimal digits. If digits are displayed, the value shown
will round from the actual value. To ensure that a point is actually at an integer value rather than a
rounded decimal value, do the following:

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1. Move the cursor over the coordinate value so it is highlighted.

2. Press + to display additional decimal digits or - to hide digits.

• The ideas contained in the following pages are intended to provide a framework as to how the activity
will progress.

Software/Detail? id=258#.

Associated Materials

• Student Worksheet: Midpoints in the Coordinate Plane http://www.ckl2.org/flexr/chapter/
9686

• Cabri Jr. Application

Problem 1 — Midpoints of Horizontal or Vertical Segments

Step 1:

• Students should open a new Cabri Jr. file. If the axes are not currently showing, they should select
Hide/Show > Axes.

They will construct a horizontal segment in the first quadrant using the Segment tool.
Step 2:

• Students will select Coord. & Eq. and show the coordinates for the endpoints of the segment.

If the coordinates of the endpoints are not integers, they need to use the Hand tool to drag the endpoints
until the coordinates are integers.

Step 3:

Students should now make a prediction about the coordinates for the midpoint of the segment.

To check their predictions, students will select Midpoint, construct the midpoint of the segment,

and then show its coordinates.

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m m m

D D P

Ci,i> CH/i)

Step 4:

Before moving on, students need to hide the coordinates of the midpoint with the Hide/Show >
Object tool.

They should use the Hand tool to drag the segment to another location. If you drag the entire
segment, it will remain horizontal.

3 www.ckl2.org

Students can make a prediction about the new coordinates of the midpoint and check their prediction by
showing the coordinates of the midpoint.

Step 5:

• Repeat this exploration with a new segment. Use a vertical segment.

If desired, have students explore midpoints of segments whose endpoints do not have integer coordinates,
or are not in Quadrant 1.

Problem 2 — Midpoints of Diagonal Segments

Step 1:

• Instruct students to open a new Cabri Jr. file. If needed, select Hide/Show > Axes to show the
coordinate axes.

• They should begin by using the Segment tool to construct a diagonal segment in the first quadrant.

Step 2:

• Students need to select Coord. & Eq. and show the coordinates for the endpoints of the segment.

If the coordinates of the endpoints are not integers, they should use the Hand tool to drag the endpoints
to make the coordinates integers.

Step 3:

• Students should now make a prediction about the coordinates for the midpoint of the segment.

To check their predictions, students will construct the midpoint of the segment and show the coordinates
of the midpoint.

Step 4:

• Students should now hide the coordinates of the midpoint with the Hide/Show > Object tool.

• Using the Hand Tool, students should drag the segment to another location. If the entire segment
is selected, it will keep the same diagonal slant.

Students will make a prediction about the new coordinates of the midpoint and check their prediction by
showing the coordinates of the midpoint.

Step 5:

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• Repeat this exploration with a new segment.

If desired, direct students to again consider segments whose endpoints do not have integer coordinates or

Step 6:

• In pairs, students should discuss the following topic:

Describe in words how to find the coordinates of the midpoint of a segment if you know the coordinates of
the endpoints. Try to write a formula or a rule for midpoints.

This activity is intended to supplement Geometry, Chapter 1, Lesson 5.

ID: 10893

Time required: 15 minutes

Activity Overview

In this introductory or review activity, students will explore vertical and adjacent angles. They will define
and identify pairs of angles. Then they will change the intersecting lines of a geometric model to make
conjectures about the relationships of the pairs of angles.

Topic: Points, Lines & Planes

• Congruency of vertical angles

• Adjacent angles formed by two intersecting lines are supplementary.

Teacher Preparation and Notes

• This activity was written to be explored with Cabri Jr.

Software/Detail? id=258#.

US/ 'Activities/Detail? id=10893 and select VERTICAL. 8xv.

Associated Materials

• Student Worksheet: Vertical and Adjacent Angles http://www.ckl2.org/flexr/chapter/9686,
scroll down to the second activity.

• Cabri Jr. Application
. VERTICAL. 8xv

Problem 1 — Exploring Vertical Angles

Students should first define the term vertical angles using their textbook or other source. After opening
Cabri Jr., student should press Y = and select Open and then VERTICAL to view the file. Introduce the
geometric model (two intersecting lines).

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Students should name the two pairs of vertical angles of the model.

Direct students to explore the model independently, by grabbing and dragging points B and/or C. To grab
a point, move the cursor over the point and then press a. The cursor will change to a closed fist. Press a
again to release the point.

Then they need to answer the questions on the worksheet.

MHQE: 150*

HCDEi 150*

HE:Q>: £0*

~"' A

M^

--"

Problem 2 — Exploring Adjacent Angles

Students are to repeat the steps from Problem 1 with adjacent angles. They will need to first define the
term adjacent angles using a textbook or other source.

Students are to use the same geometric model of intersecting lines from Problem 1. They need to name
the four pairs of adjacent angles.

Have them explore the model independently and make a conjecture about adjacent angles.

They can also use the Calculate tool from the F5 menu to add the pairs of angles confirming that adjacent
angles are supplementary. To do this, select one angle measurement, press +, and then select the adjacent
angle measurement.

Solutions

1. Two angles whose sides are opposite rays.

2. IAOB and ICOD; IBOC and lAOD

Table 2.1:

Location I s

-ynd ord Ath

mlAOB

130.6

mlBOC

49.4

mlCOD

130.6

mlAOD

49.4

118.5 90.4 79.4

61.5 89.6 110.6

118.5 90.4 79.4

61.5 89.6 100.6

4. If lAOD and IBOC are vertical angles, then the mlAOD = mlBOC.

5. If lAOB and ICOD are vertical angles, then the mlAOB = mlCOD.

6. Vertical angles are congruent.

7. Adjacent angles are two coplanar angles that have a common side and a common vertex but no common
interior points.

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8. IAOB and IBOC; IBOC and ICOD; LAOD and LCOD; LAOD and lAOB

9. Adjacent angles formed by two intersecting lines are supplementary.

10. If lAOB andlBOC are adjacent angles formed by two intersecting lines, then LAOB and IBOC are
supplementary.

11.

3x = 75
x = 25

y = 180 - 75 = 105

12.

x + 10 = 4x - 35

3x = 45

x= 15

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Chapter 3

TE Reasoning and Proof - TI

3.1 Conditional Statements

This activity is intended to supplement Geometry, Chapter 2, Lesson 2.

ID: 8746

Time required: 40 minutes

Activity Overview

In this activity, students will write logical statements related to the given conditional statement. They will
explore whether the statements are true or false and find counterexamples for false statements. These explo-
rations will involve the slopes of parallel and perpendicular lines and lengths of collinear and noncollinear
segments.

Topic: Inductive & Deductive Reasoning

• Write the inverse, converse, and contrapositive statements corresponding to a given conditional state-
ment.

• Use a counterexample to prove that a statement is false.

Teacher Preparation and Notes

• This activity is designed to be used in a high school or middle school geometry classroom.

• Before beginning this activity, students should be familiar with the terms inverse, converse, and
contrapositive.

• Students will discover the following concepts:

— Parallel lines have slopes that are equal; perpendicular lines have slopes that are opposite recip-
rocals (the product of the slopes is -1).

— The Segment Addition Postulate states that AB + BC = AC if B is between A and C and the
points are collinear. If the points are not collinear, then AB + BC > AC.

• If desired, teachers can explore which of the statements in the activity are also biconditional statements
(definitions that are always true).

• This activity is designed to be student- centered with the teacher acting as a facilitator while students
work cooperatively. Use the following pages as a framework as to how the activity will progress.

www.ckl2.org 8

Software/Detail ?id=258#.

US/Activities/Detail?id=8746 and select COLSEG.8xv and NOCOLSEG.8xv.

Associated Materials

• Student Worksheet: Conditional Statements http://www.ckl2.org/flexr/chapter/9687

• Cabri Jr. Application

. C0LSEG.8xv and N0C0LSEG.8xv

Problem 1 — Slopes of lines

To begin, students should open a new Cabri Jr. file.

Step 1: Students will need to construct two parallel lines. First, a line needs to be constructed using the
Line tool and a point not on the line using the Point tool.

Step 2: Using the Parallel tool, a line parallel to the existing line should be constructed through the
point not on the original line.

Step 3: Students will find the slope of both lines by using the Slope tool (Measure > Slope). Students
can now use the Hand tool to drag the original line or the point and observe the results.

What is true of the slopes of parallel lines?

Students should record their observations on the worksheet part A. They will write the converse, inverse,
and contrapositive of the statement and determine the truth of each.

Students will next construct perpendicular lines in a new Cabri Jr. file.

Step 4: Again construct a line and a point not on the line. Have students select the Perp. tool to
construct a line perpendicular to the existing line through the point not on the original line.

Step 5: Students will find the slope of both lines. They should drag the original line or the point and
observe the results.

Step 6: Have students select Calculate to find the product of the slopes. They should again drag the
line to observe the results.

9 www.ckl2.org

What is true of the slopes of perpendicular lines?

Students should record their observations, write conditional statements, and determine the truth of the
statements on the worksheet in part B.

On a new Cabri Jr. file, students will construct two lines that have the same y-intercept. If the axes are
not currently showing, they should select Hide/Show > Axes.

Step 7: Students should start by placing a point on the y-axis using the Point > Point On tool.

Step 8: Make sure that students construct two different lines with the same y-intercept (the point
previously created on the y-axis) and find the slopes of both lines.

Note: If desired, students can display the equations of both lines using the Coord. & Eq. tool instead of
the slopes.

Step 9: Direct students to drag the lines and the y-intercept point and observe the changes in the
coordinates and slopes.

Students will write conditional statements on the worksheet part C. They should sketch counterexamples
for any false statements.

The screenshot at right shows a counterexample for the converse statement "If two lines have different
slopes, then they have the same y— intercept."

Problem 2 — Collinear and noncollinear segments

Step 1: Distribute the file COLSEG to students. Points A, B, and C are constructed on a line and the
lengths AB, BC, and AC are displayed.

Ht-Z.Z IK.-I.B

Step 2: Students will drag the points with the Hand tool and observe the changes in the lengths.

Be sure that students drag all three points, and change the order of the points (they should investigate
situations where B is not between the other two points).

Students should record their observations on the worksheet and write a conditional statement.

Step 3: Distribute the file NOCOLSEG to students. Points A, B, and C are not collinear. The lengths
AB, BC, and AC are displayed.

Step 4: Students should now drag the points and observe the changes in the distances. If desired, students
can use the Calculate tool to find the sum of AS + BC.

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Students should record observations on the worksheet and write a conditional statement.

Solutions

Problem 1

A. If two lines are parallel, then the slopes of the lines are equal.
Converse: If the slopes of the lines are equal, then the two lines are parallel.
Inverse: If the two lines are not parallel, then the slopes of the lines are not equal.
Contrapositive: If the slopes of the lines are not equal, then the two lines are not parallel.
Each conditional statement is true.

B. If two lines are perpendicular, then the slopes of the lines are equal to -1.
Converse: If the slopes of two lines are equal to -1, then the lines are perpendicular.
Inverse: If two lines are not perpendicular, then the slopes of the lines are not equal to -1.
Contrapositive: If the slopes of two lines are not equal to -1, then the lines are not perpendicular.
Each conditional statement is true.

C. Converse: If two lines have different slopes, then the lines have the same v-intercept.

Inverse: If two different lines do not have the same v-intercept, then the lines do not have different slopes.

Contrapositive: If two lines do not have different slopes, then the different lines do not have the same
v-intercept.

Each conditional statement is false.

Problem 2

A. Answers will vary for distances of AB, BC, and AC.

The lengths AB and BC add up to equal AC when B is between A and C.

If A, B, and C are collinear and B is between A and C, then AB to +BC = AC.

B. Answers will vary for distances of AB, BC, and AC.

AB + BC> AC

If A, B, and C are not collinear, then the sum of the lengths of AB and BC is greater than AC.

11 www.ckl2.org

Chapter 4

TE Parallel and
Perpendiculuar Lines - TI

4.1 Parallel Lines cut by a Transversal

This activity is intended to supplement Geometry, Chapter 3, Lesson 2.

ID: 8681

Time required: 30 minutes

Topic: Points, Lines & Planes

• Construct two parallel lines and an intersecting transversal and measure the angles to discover that
pairs of alternate angles (interior and exterior) are equal and same side angles (interior and exterior)
are supplementary.

Activity Overview

In this activity, students will develop and strengthen their knowledge about the angles formed when parallel
lines are cut by a transversal. The measurement tool helps them discover the relationships between and
among the various angles in the figure. Students will draw parallel lines, draw a transversal, and explore
angle relationships.

Teacher Preparation

• This activity is designed to be used in a high-school geometry classroom. It can also be used with

• Students should already be familiar with parallel lines as well as reading and naming angles.

• The screenshots on pages 1-3 demonstrate expected student results.

Classroom Management

• This activity is intended to be mainly teacher-led, with breaks for individual student work. Use the
following pages to present the material to the class and encourage discussion. Students will follow
along using their calculators.

www.ckl2.org 12

Software/Detail ?id=258#.

Associated Materials

• Student Worksheet: Parallel Lines cut by a Transversal http://www.ckl2.org/flexr/chapter/
9688

• Cabri Jr. Application

See the student worksheet on how to complete this activity.

4.2 Transversals

This activity is intended to supplement Geometry, Chapter 3, Lesson 3.

ID: 10989

Time required: 15 minutes

Activity Overview

In this activity, students will explore corresponding, alternate interior and same-side interior angles. This
is an introductory activity where students will need to know how to grab points in Cabri Jr.

Topic: Points, Lines & Planes

• Corresponding angles are congruent

• Alternate Interior angles are congruent

• Same-Side Interior angles are supplementary

Teacher Preparation and Notes

• This activity was written to be explored with Cabri Jr.

Software/Detail? id=258#.

US/Activities/Detail?id=10989 and select TRNSVRSL.8xv.

Associated Materials

• Student Worksheet: Transversals http://www.ckl2.org/flexr/chapter/9688, scroll down to the
second activity.

• Cabri Jr. Application
. TRNSVRSL.8xv

Exploring Parallel Lines cut by a Transversal

Before beginning the activity, students should know the definition of corresponding, alternate interior and
same-side interior angles. Questions 1, 2, and 3 ask students to name pairs of angles from the diagram.
This should be done without the use of the calculator.

13 www.ckl2.org

-oE--

Students will now run the Cabri Jr. App and open the file TRNSVRSL. To open a file, they should press
Y = and select Open.

By moving point G, students will discover the properties of two parallel lines cut by a transversal. To move
a point, students need to move the cursor over the point (a square) and press ALPHA.

For Questions 4, 5, and 6, students will move point G to four different places. They should record the
angle measurements in the tables on the worksheet. Then, students should try to generalize their results
in the Conjecture section.

There are two application problems at the end of the worksheet for students to apply what they have
learned in the activity. These problems can be done as homework.

Solutions

1. Z4 and Z5 is another pair

2. Z4 and Z6 is another pair

3. Z4 and Z8 is another pair

4. a. Corresponding

b. Sample measurements.

Table 4.1:

1 st position

2 position

3 position

4 position

mlABC
mlHFB

109
109

84
84

56
56

37
37

c. Congruent

5. a. Same-Side Interior

b. Sample measurements.

Table 4.2:

I st position

2 position

3 position

4 position

mlABF
mlHFB

150
30

136
44

112
68

75
105

c. Supplementary

6. a. Alternate Interior

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14

b. Sample measurements.

Table 4.3:

1 st position

2 position

3 position

4 position

mlDBF
mlBFH

43

43

60
60

108
108

125
125

c. Congruent
Conjectures

For parallel lines and a transversal...

7. if two angles are corresponding angles, then they are congruent.

8. if two angles are alternate interior angles, then they are congruent.

9. if two angles are same-side interior angles, then they are supplementary.
Extra Problems

10. Zl, Z2, and Z3 are all equal to 55°
11.

108 = 7x - 4 and y
112 = 7x
16 = x

72

4.3 Perpendicular Slopes

This activity is intended to supplement Geometry, Chapter 3, Lesson 4-

ID: 8973

Time required: 45 minutes

Topic: Linear Functions

• Graph lines whose slopes are negative reciprocals and measure the angles to verify they are perpen-
dicular.

Activity Overview

In this activity, students investigate the "negative reciprocal" relationship between the slopes of perpendicular
lines. The final phase of the activity is appropriate for more advanced students as they are led through an
algebraic proof of the relationship. Optional geometric activities (Problems 5 and 6) use the result to verify
that (1) the radius of a circle and its tangent line are perpendicular and (2) a triangle inscribed in a circle
with the diameter as one side is a right triangle.

Teacher Preparation

This activity is appropriate for students in Algebra 1. It is assumed that students have recently been
introduced to the notion of slope and perhaps the fact that two lines are parallel if and only if they have the
same slope.

15

www.ckl2.org

Classroom Management

• This activity is designed to have students explore individually and in pairs. However, an alternate
approach would be to use the activity in a whole-class format. By using the computer software and
the questions found on the student worksheet, you can lead an interactive class discussion on the
slope of perpendicular lines.

• This worksheet is intended to guide students through the main ideas of the activity. You may wish
to have the class record their answers on a separate sheet of paper, or just use the questions posed to
engage a class discussion.

Software/Detail? id=258#.

US/Activities/Detail?id=8973 and select PERP1-PERP6

Associated Materials

• Student Worksheet: Perpendicular Slopes http : //www . ckl2 . org/f lexr/chapter/9688, scroll down
to the third activity.

• Cabri Jr. Application

. PERP1.8xv, PERP2.8xv, PERP3.8xv, PERP4.8xv, PERP5.8xv, and PERP6.8xv

See the student worksheet on how to complete this activity.

www.ckl2.org 16

Chapter 5

TE Congruent Triangles - TI

5.1 Interior and Exterior Angles of a Triangle

This activity is intended to supplement Geometry, Chapter 4; Lesson 1.

ID: 8775

Time required: 40 minutes

Activity Overview

In this activity, students will measure interior and exterior angles of a triangle and make conjectures about
their relationships.

Topic: Triangles & Congruence

• Use inductive reasoning to conjecture a theorem about the total measures of a triangle's interior
angles.

• Prove that the sum of the measures of the interior angles of a triangle is 180°.

• Prove that the sum of the measures of the exterior angles of a triangle is 360°.

Teacher Preparation

• This activity is designed to be used in a high school or middle school geometry classroom.

• This activity is designed to be student- centered with the teacher acting as a facilitator while students
work cooperatively. Use the following pages as a framework as to how the activity will progress.

• If needed, review with students the definitions of the following angles before the activity: Interior
angle, exterior angle, remote interior angle, adjacent interior angle.

• Note: Measurements can display 0, 1, or 2 decimal digits. If digits are displayed, the value shown
will round from the actual value. To change the number of digits displayed:

1. Move the cursor over the coordinate value so it is highlighted.

2. Press + to display additional decimal digits or - to hide digits.

Software/Detail? id=258#.

17 www.ckl2.org

Associated Materials

• Student Worksheet: Interior and Exterior Angles http://www.ckl2.org/flexr/chapter/9689.

• Cabri Jr. Application

Problem 1 — Interior angles of a triangle

Students should open a new Cabri Jr. file. They will construct a triangle using the Triangle tool.

A

^ i

Select the Alph-Num tool to label the vertices A, B, and C as shown.

Note: Press ENTER to start the label, then press ENTER again to end the label.

Have students measure the three interior angles of the triangle using the Measure > Angle tool.

Note: To measure an angle, press ENTER on three points, with the vertex of the angle being the second
point selected.

Students should records this data in the first row of the chart on the student worksheet.

Students should drag a vertex of the triangle to change the angle measures. Have them try to create
different types of triangles (acute, obtuse, right) and record two more sets of data in the chart.

From here, they should make a conjecture about the three interior angles.

,-**

■"■ 55^ j^

i c

Instruct students to use the Calculate tool to find the sum of the three interior angles of the triangle.
Drag a vertex and observe the results. Ask: Do the results support your conjecture?
Have students save the file as IntAngle.

Problem 2 — One exterior angle of a triangle

Students should continue using the previous file, but instruct them to save the file as ExtAngle using the
Save As option.

Have students construct a line through the two lower vertices of the triangle (A and C) using the Line
tool.

Note: To be certain that the line passes through a vertex, be sure that the vertex point is flashing before
pressing ENTER.

www.ckl2.org 18

Next, students should create a new point on the line to the right of the triangle using the Point > Point
On tool. Label it D.

A

/ \

/'H7» 7i-> ^

"k

Direct students to measure the exterior angle IBCD with the Measure > Angle tool. Record this measure
and the measures of the interior angles on the chart on the student worksheet.

They can then drag a vertex of the triangle to change the angle measures, and add two more sets of data
to the chart.

Students should now make some observations about the exterior angle and its relationship to other angles
in the chart.

Encourage them to make calculations as needed to test their conjectures. If needed, suggest that they
calculate the sums of pairs of angles in the chart.

Again, students can drag a vertex and observe the results. Ask: Do the results support your conjectures?

Before proceeding, have students save the file as ExtAngle.

Problem 3 — Three exterior angles of a triangle

Students should continue using the previous file, but instruct them to save the file as ExtAng3 using the
Save As tool.

Note: Press the ALPHA button to access the numeric character "3."

Students will select the Line tool and construct a line through vertices A and B of the triangle as shown
below.

Next, students should measure one exterior angle at each vertex.

Note: It is not necessary to create an additional point on the line before measuring the angle.

y

M 117*

/A

/ \

_i33y nr» ?±> nag* _

DP D D —

C [■

7*

Have them record the measures of the three exterior angles into the chart on the student worksheet.

They can then drag a vertex of the triangle and record more data into the chart.

Instruct students to make a conjecture about the three exterior angles, and have them calculate the sum
of the three exterior angles.

Tell them to drag a vertex and observe the results. Ask: Do the results support your conjectures?

19 www.ckl2.org

Solutions

1. IB = 43°, IBCD = 143°

2. IBCA = 117°,Z5 = 32°

3. IBCA = 81°,int lA = 42°,ext lA = 138°

4. IBCA = 83°,ext Z/4 = 140°, lABC = 57°,ext Zfl = 123°

5.2 Congruent Triangles

This activity is intended to supplement Geometry, Chapter 4> Lesson 4-

ID: 8817

Time required: 45 minutes

Topic: Triangles & Congruence

• Investigate the SSS, SAS, and ASA sets of conditions for congruent triangles.

Activity Overview

In this activity, students will explore the results when a new triangle is created from an original triangle
using the SSS, SAS, and ASA sets of conditions for congruence. In doing so, they will use the Cabri Jr.
Compass tool to copy a segment and the Rotation tool to copy an angle.

Teacher Preparation

• This activity is designed to be used in a high school or middle school geometry classroom.

• The Compass tool copies a given length as a radius of a circle to a new location.

• The screenshots on pages 1-1 demonstrate expected student results.

Classroom Management

• This activity is designed to be student- centered with the teacher acting as a facilitator while students
work cooperatively. Use the following pages as a framework as to how the activity will progress. It is
recommended that you print out the following pages as a reference for students.

• If desired, have students work in groups with each group focusing on one of the problems.

• The student worksheet helps guide students through the activity and provides a place for students to

• The student worksheet includes an optional extension problem to investigate two corresponding sides
and the NON-included angle.

Software/Detail? id=258#.

Associated Materials

• Student Worksheet: Congruent Triangles http://www.ckl2.org/flexr/chapter/9689, scroll down
to the second activity.

• Cabri Jr. Application

www.ckl2.org 20

Problem 1 — Three Corresponding Sides (SSS)

Step 1: Students should open a new Cabri Jr. file.

They will first construct a scalene triangle using the Triangle tool.

Step 2: Select the Alph-Num tool to label the vertices A, B, and C as shown.

/

Step 3: Have students construct a free point on the screen using the Point tool. Label it point D with
the Alph-Num tool.

Step 4: Students will select the Compass tool to copy AB to point D.

• Press ENTER on AB. A dashed circle will appear and follow the pointer.

• Press ENTER on point D. The compass circle is anchored at center D.

Have students construct a point on the compass circle and label it point E.

f

/

,'

s~~ ~~\

p

k ,. r

'■■--_---' fc

Step 5: Direct students to create DE with the segment tool. This segment is a copy of AB.

Hide the compass circle with the Hide/Show > Object tool.

Save this file as CongTri. This setup will be used again for Problems 2 and 3.

Step 6: Drag point E, and observe that the location of DE can change but it will not change in length.
You must drag either point A or point B to change the length of DE.

Step 7: To copy the other two segments of AABC, repeat the use of the Compass tool.

Students should copy AC to point D.

Students should copy BC to point E.

Find the intersection point of the two compass circles and hide the circles. Label the intersection point as
F.

Finally, construct segments DF and EF to complete the new triangle.

21

www.ckl2.org

Step 8: Students will now investigate whether ADEF is congruent to AABC.

Measure the sides and angles of both triangles to confirm that all corresponding parts are congruent. Use
the Measure > D. & Length and Measure > Angle tools.

Step 9: Drag vertices of the triangles and observe the results.

Dragging vertices of AABC will cause both triangles to change in size and shape.

Dragging vertices of ADEF will cause ADEF to change its location but its size and shape will remain the
same as AABC. Notice that not all vertices of ADEF are available to be dragged, since they have been
constructed to be dependent on AABC.

If students wish to save their work, use the Save As tool and do not reuse the name CongTri.

Problem 2 — Two Corresponding Sides and the Included Angle

(SAS)

Step 1: Students should open the Cabri Jr. file CongTri that they created in Problem 1.

Recall that DE is a copy of AB.

Step 2: Students will select the Rotation tool to copy lABC.

• Press ENTER on point E. This is the center of rotation.

• Press ENTER on DE. This is the object to be rotated.

• Press ENTER three times on the vertices A, B, and C in that order to identify the angle of rotation.
A new rotated segment appears.

/

/f

J

E
k

Step 3: Have students use the Line tool to construct a line from point E through the endpoint of the
new segment.

Use the Hide/Show > Object tool to hide the endpoint of the rotated segment.

Step 4: Use the Compass tool to copy BC to point E.

Create the intersection point of the compass circle and the line and label it point F .

Hide the compass circle and the line.

Finally, construct DF and EF to complete the new triangle.

www.ckl2.org 22

I\

/

™ .

'* 1 [.^""

iE

1

*

Step 5: Students will now investigate whether ADEF is congruent to AABC.

Measure the sides and angles of both triangles to confirm that all corresponding parts are congruent.

Drag vertices of the triangles and observe the results.

If students wish to save their work, use the Save As tool and do not reuse the name CongTri.

Problem 3 — Two Corresponding Angles and the Included Side

(ASA)

Step 1: Students should open the Cabri Jr. file CongTri that they created in Problem 1.

Recall that DE is a copy of AB.

Step 2: Students will select the Rotation tool to copy lABC.

• Press ENTER on point E. This is the center of rotation.
. Press ENTER on ~DE. This is the object to be rotated.

Press ENTER three times on the vertices A, B, and C in that order to identify the angle of rotation. A
new rotated segment appears.

Step 3: Students will select the Rotation tool to copy IBAC.

• Press ENTER on point D. This is the center of rotation.

• Press ENTER on DE. This is the object to be rotated.

• Press ENTER three times on the vertices B, A, and C in that order to identify the angle of rotation.
A new rotated segment appears.

?' :

I

/J,

Step 4: Have students use the Line tool to construct lines over the new segments.
Use the Hide/Show > Object tool to hide the endpoints of the rotated segments.

23

www.ckl2.org

Step 5: Create the intersection point of the two lines and label it point F. Hide the lines.

Finally, construct DF and EF to complete the new triangle.

Step 6: Students will now investigate whether ADEF is congruent to AABC.

Measure the sides and angles of both triangles to confirm that all corresponding parts are congruent.

Drag vertices of the triangles and observe the results.

If students wish to save their work, use the Save As tool and do not reuse the name CongTri.

5.3 Triangle Sides and Angles

ID: 8795

Time required: 40 minutes

Topic: Triangles &: Congruence

• Classify triangles by angle measure.

• Prove and apply the Isosceles Triangle Theorem.

• Prove and apply the converse of the Isosceles Triangle Theorem.

• Recognize the relationship between the side lengths and angle measures of a triangle.

Activity Overview

In this activity, students will explore side and angle relationships in a triangle. First, students will discover
where the longest (and shortest) side is located relative to the largest (and smallest) angle. Then, students
will explore the Isosceles Triangle Theorem and its converse. Finally, students will determine the number
of acute, right, or obtuse angles that can exist in any one triangle.

Teacher Preparation

• This activity is designed to be used in a high school or middle school geometry classroom.

• The screenshots on pages 1-3 demonstrate expected student results.

Classroom Management

• This activity is designed to be student- centered with the teacher acting as a facilitator while students
work cooperatively. Use the following pages as a framework as to how the activity will progress.

• The student worksheet helps guide students through the activity and provides a place for students to

• Note: Measurements can display 0, 1, or 2 decimal digits. If digits are displayed, the value shown
will round from the actual value. To change the number of digits displayed:

www.ckl2.org 24

1. Move the cursor over the coordinate value so it is highlighted.

2. Press + to display additional decimal digits or - to hide digits.

Software/Detail? id=258#.

Associated Materials

• Student Worksheet: Triangle Sides and Angles http://www.ckl2.org/flexr/chapter/9689, scroll
down to the third activity.

• Cabri Jr. Application

Problem 1 — Size and Location of Sides and Angles

Step 1: Opening a new Cabri Jr. file, students should first construct a triangle using the Triangle tool.

Step 2: Select the Alph-Num tool to label the vertices A, B, and C as shown.

Note: Press ENTER to start entering the label, and then press ENTER again to end the label.

A

*£ h

H C

Step 3: Students should measure the three interior angles of the triangle using the Measure > Angle
tool. To measure an angle, press ENTER on each of three points that define the angle, with the vertex
of the angle as the second point chosen.

Step 4: Direct students to measure the three side lengths using the Measure > D. & Length tool.

Note: To measure a side of the triangle, you must click on the endpoints of the segment. If you click on
the side itself, the tool will return the perimeter of the triangle.

Step 5: Now students should drag a vertex of the triangle to change the angle measures and side lengths.

Encourage them to make a conjecture about the sizes and locations of the angles and sides in a triangle.

Students can answer the question on the worksheet based on their observations, concluding that the largest
angle is opposite the longest side, and the smallest angle is opposite the shortest side.

Step 6: Have students save this file as Triangle.

Problem 2 —The Isosceles Triangle Theorem

Step 1: Distribute the Cabri Jr. files Isostril and Isostri2 to student calculators

Have students open the file Isostril. An isosceles triangle has been constructed and the congruent sides
have side lengths displayed.

Step 2: Students should measure all three angles using the Measure > Angle tool.

Step 3: Direct students to drag a vertex of the triangle to explore what happens to the angle measures
when two sides have equal lengths.

25 www.ckl2.org

Students should conclude that a triangle with two congruent sides also has two congruent angles. This is
known as the Isosceles Triangle Theorem.

Step 4: The converse of the Isosceles Triangle Theorem may be explored in the file Isostri2.

Step 5: Students should measure all three sides using the Measure > Length tool, then drag a vertex
to explore.

Step 6: On their worksheets, students should make the conclusion that a triangle with two congruent
angles also has two congruent sides.

Problem 3 — Types of Angles in a Triangle

Step 1: Tell students to open the file Triangle that they saved from Problem 1.

If desired, have them hide the side lengths using the Hide/Show > Object tool.

Step 2: Students should drag a vertex and notice how many angles of each type can exist in a triangle.
They should conclude that a triangle:

• can have three acute angles

• cannot have three right angles

• cannot have three obtuse angles

Challenge them to observe that a triangle can have, at most, one right or one obtuse angle and that a
triangle cannot have both a right and obtuse angle.

www.ckl2.org 26

Chapter 6

TE Relationships within
Triangles - TI

6.1 Perpendicular Bisector

This activity is intended to supplement Geometry, Chapter 5, Lesson 2.

ID: 11198

Time required: 15 minutes

Activity Overview

In this activity, students will explore the perpendicular bisector theorem and discover that if a point is
on the perpendicular bisector of a segment, then the point is equidistant from the endpoints. This is an
introductory activity where students will need to know how to grab and move points, measure lengths, and
construct the perpendicular bisector with Cabri Jr.

Topic: Triangles &: Their Centers

• Perpendicular Bisector Theorem
Teacher Preparation and Notes

• This activity was written to be explored with the Cabri Jr. on the TI-84-

Software/Detail? id=258#.

US/Activities/Detail?id=11198 and select PERPBIS and POINTS.

Associated Materials

• Student Worksheet: Perpendicular Bisectors http://www.ckl2.org/flexr/chapter/9690

• Cabri Jr. Application

. PERPBIS.8xv and POINTS.8xv

27 www.ckl2.org

In this activity students will explore the properties of perpendicular bisectors. In the first part, students
will measure the distance to a point on the bisector to identify congruent segments. In the second part,
students will apply what they have learned about perpendicular bisectors to the context of map coordinates.

Problem 1 — Exploring the Perpendicular Bisector Theorem

Students will be exploring the distance from a point on the perpendicular bisector to the endpoints of a
segment and discover that if a point is on the perpendicular bisector of a segment, then it is equidistant
from the two endpoints of the segment.

Students will measure the lengths of segments using the Distance & Length tool (press GRAPH and
select Measure > D.&Length). To drag a point, students will move the cursor over the point, press
ALPHA, move the point to the desired location, and then press ALPHA again to release the point.

>

Is \ k

i y

A rf- \

>,

Problem 2 — An Application of the Perpendicular Bisector The-
orem

Students will apply what they learned from problem one and find the intersection of two perpendicular
bisectors to find a point equidistant from three points.

To draw a segment, students should use the Segment tool (press WINDOW and select Segment). To
draw a perpendicular bisector, press ZOOM and select Perp. Bis..

I.K..VJ

Cl/J.53q V C5*»

S&

Solutions

Problem 1 — Exploring the Perpendicular Bisector Theorem

• Answers will vary. The measures of AC and BC should be equal.

• Sample answer: AC is congruent to BC. The distances are the same.

• If a point is on the perpendicular bisector of a segment, then it is equidistant from the two endpoints
of the segment.

Problem 2 — An Application of the Perpendicular Bisector Theorem

www.ckl2.org 28

(6,7)
Fl

6.2 Hanging with the Incenter

This activity is intended to supplement Geometry, Chapter 5, Lesson 3.

ID: 11360

Time Required: 45 minutes

Activity Overview

In this activity, students will explore the angle bisector theorem and discover that if a point is on the angle
bisector of a segment, then the point is equidistant from the sides. This is an introductory activity, but
students will need to know how to grab and move points, measure lengths, and construct the perpendicular
bisector with Cabri Jr.

Topic: Triangles &: Their Centers

• Angle Bisector Theorem

• Incenter

Teacher Preparation and Notes

• This activity was written to be explored with the Cabri Jr. on the TI-84-

Software/Detail? id=258#.

US/ 'Activities/Detail? id=11360 and select ANGBIS.

Associated Materials

• Student Worksheet: Hanging with the Incenter http://www.ckl2.org/flexr/chapter/9690, scroll
down to the second activity.

• Cabri Jr. Application
. ANGBIS.8xv

Problem 1 — Exploring the Angle Bisector Theorem

Students will be exploring the distance from a point on the angle bisector to the segment. They will
discover that if a point is on the angle bisector of an angle, then it is equidistant from the two segments of
the angle.

Students will measure the lengths of segments using the Distance & Length tool (press GRAPH and
select Measure > D.& Length). To drag a point, students will move the cursor over the point, press
ALPHA, move the point to the desired location, and then press ALPHA again to release the point.

29 www.ckl2.org

Problem 2 — Exploring the Incenter of a Triangle

Students will need to open a new Cabri Jr. file by pressing Y =, selecting New, and answer no if asked
to save.

Students are to create an acute triangle and find the angle bisector of all three angles of the triangle.
Students should realize that they are concurrent and answer Questions 4-8 on the accompanying worksheet.

Students will need to find the distance from the incenter to the 3 sides of the triangle. Students may need
to be reminded how to find the distance from a point to a line or segment. It is the perpendicular distance
from the incenter to the side. Teachers should encourage the students to hide their angle bisectors. Be
sure that all students understand that the perpendicular line created from the incenter to a side of the
triangle is not the perpendicular bisector.

Students should discover that all of these distances are equal.

Problem 3 — Extension

Problem 3 is an extension of this activity. Students will use the handheld and the angle bisectors to find
the coordinates of fence posts on a plot of land. Note: This problem will be very hard for the average
student.

Students must first set up coordinates for the 2 fences at (-1, 2), (1, 0), (1, 3), and (5, 3). Next, students
should create segments for the two fences. From here, there are several approaches to solving this problem.

Using the incenter:

One approach is to find the incenter of the triangle formed by the intersection of the two fences (over the
pond) and a segment joining the two fences. Students should construct a segment from (1, 0) to (5, 3)
connecting the two fences. This is to create two angles that they can use to find angle bisectors. The point
of intersection of the two angle bisectors is the incenter of the triangle. This point is therefore equidistant
from the two fences.

However, we need two points to determine the line where the fence will be. Students will need to create a
similar triangle that connects the two fences. To guarantee similarity, choose a segment that is parallel to
the segment joining (1, 0) and (5, 3) using the Parallel tool (Press ZOOM, then scroll down to Parallel).
Find the incenter of the second triangle and you can create the line through those two incenters. One point
on the line should be at (5.24, 0) or (5.2, 0) and the other can be any other point on the line connecting
the two incenters. The y-intercept is another convenient point to find the coordinates of (0, 2.2).

Using only an angle bisector:

Another approach is to find the point of intersection of the fences (if they were continued over the pond) .
Students can construct lines for the Fence 1 and Fence 2 and plot the intersection point of the lines.
Then, students can use the Angle Bisector tool to find the angle bisector of this angle. This ray will be
equidistant from Fence 1 and Fence 2. Students can plot a point on this ray and display and record the
coordinates of this point. Then, move the point and record the new coordinates.

www.ckl2.org 30

i

'.l.v/sMS.ifjs.v

K

\

Solutions

Table 6.1:

Position

1 st position

2 nd position

3 rd position

4 th position

DF
DE

2.25473
2.25473

2.72284
2.72284

1.85757
1,85757

1.12521
1.12521

2. They are equal.

3. Equidistant

4. They are concurrent.

5. Not possible

6. Not possible

7. Obtuse, Acute, and Right

8. They are all equal.

9. Sample answer: (5.2, 0) and (0, 2.2)

6.3 Balancing Point

This activity is intended to supplement Geometry, Chapter 5, Lesson 4-

ID: 11403

Time Required: 45 minutes

Activity Overview

In this activity, students will explore the median and the centroid of a triangle. Students will discover
that the medians of a triangle are concurrent. The point of concurrency is the centroid. Students should
discover that a triangle's center of mass and centroid are the same.

Topic: Triangles &: Their Centers

• medians

• centroid

Teacher Preparation and Notes

31

www.ckl2.org

This activity was written to be explored with the TI-84 using the Cabri Jr. application.

This is an introductory activity for which students will need to know how to construct triangles, grab

and move points, measure lengths, and construct segments.

Software/Detail? id=258#.

US/Activities/Detail?id=11403 and select CENTROID, MEDIAL, and MIDSEG.

Associated Materials

Student Worksheet: Balancing Point http://www.ckl2.org/flexr/chapter/9690, scroll down to

the third activity.

Cabri Jr. Application

CENTR0ID.8xv, MEDIAL.8xv, MIDSEG.8xv

Problem 1 — Exploring the Centroid of a Triangle

Students are to cut out the triangle on their worksheet and attempt to balance it with their pencil. Students
will mark the balancing point on their triangle.

Students will then use their handhelds to find the point where the triangle will be balanced. They will be
asked to compare their findings to see if their point was close to the point given by the calculator. Explain
to the students that the balancing point for an object is called the center of mass.

Problem 2 — Exploring the Medians of a Triangle

Students will need to define the terms median of a triangle and centroid from either their textbook or
another source.

Students will then create the three medians of the triangle given in Centroid. 8xv. Students will need to
find the midpoint of the three sides of a triangle. Students can access the Midpoint tool by selecting
ZOOM > Midpoint. Then, students can click on a side of the triangle to place the midpoint.

Next, students should construct the segment connecting point A and the opposite midpoint. Students can
find the Segment tool by selecting WINDOW > Segment. They should repeat this for the other two
medians.

Students are asked several questions on their accompanying worksheet. Students should be able to find
the centroid of a triangle and understand that this point is the center of mass.

I & a ™

let

i? f ^ —

www.ckl2.org 32

Problem 3 — Extending the Centroid

Students will extend the concept of the centroid in Problem 3. First, define the medial triangle to be
the triangle formed by connecting the midpoints of the sides of a triangle.

In Medial. 8 xv, students will see a triangle and its centroid. Student should construct the medial triangle
and the centroid of the medial triangle.

Students will notice that the centroid of the original triangle and the centroid of the medial triangle are
the same.

Problem 4 — Extending the Median

In Problem 4, students will extend the concept of the median. Tell students that the midsegment is a line
segment joining the midpoints of two sides of a triangle.

In Midseg.8xv, students will see AABC with midpoints D, E, and F of sides AC, AB, and BC, respectively.

Students should then construct the median that extends from point A to point F. Next, they should
construct the intersection point, G, of the median and the midsegment.

Students should find the lengths of AG, FG ,DG, and EG. Students can find the Length tool by selecting
GRAPH > Measure > D. & Length.

Students will notice that the median and midsegment bisect each other.

Solutions

1. Student solutions will vary. Students' balancing points should be close to (2, 0).

2. The median of a triangle is the segment joining the vertices of a triangle to the midpoint of the
opposite side.

3. They are concurrent.

4. (2, 0)

5. They are close.

6. (=^, =*£=*) = (2.0)

7. Centroid is the point of concurrency of the medians.

8. They are the same.

9. They bisect each other.

6.4 Hey Ortho! What's your Altitude?

This activity is intended to supplement Geometry, Chapter 5, Lesson 4-

ID: 11483

Time Required: 45 minutes

Activity Overview

In this activity, students will explore the altitudes of a triangle. Students will discover that the altitude
can be inside, outside, or a side of the triangle. Students will discover that the altitudes are concurrent.
The point of concurrency is the orthocenter. Students should discover the relationship between the type of

33 www.ckl2.org

triangle and the location of the point of concurrency. Students will discover properties of the orthocenter
in equilateral triangles.

Topic: Triangles &: Their Centers

• Altitudes of a Triangle

• Orthocenter

Teacher Preparation and Notes

• This activity was written to be explored with the Cabri Jr. on the TI-84-

Software/Detail? id=258#.

US/Activities/Detail?id=11483 and select ACUTE, EQUILATE, MEDIAL2, OBTUSE
and TRIANGLE.

Associated Materials

• Student Worksheet: Hey Ortho! Whats your Altitude? http://www.ckl2.org/flexr/chapter/
9690, scroll down to the fourth activity.

• Cabri Jr. Application

. ACUTE.8xv, OBTUSE.8xv, TRIANGLE.8xv, EQUILATE.8xv, and MEDIAL2.8xv

Problem 1 — Exploring the Altitude of a Triangle

Students should define the altitude of a triangle using their textbooks or other source on their worksheet.

In files ACUTE and OBTUSE, students are to create the altitude of an acute triangle, obtuse triangle,
and right triangle, respectively. Students will need to construct the perpendicular line to a line through a
point (press ZOOM and select Perp.).

To show the altitude of the triangle, students will need to find the intersection point of the perpendicular
line and the line that extends from the base of the triangle. Students will then need to create a segment
from the opposite vertex to the intersection point.

Problem 2 — Exploring the Orthocenter

In file TRIANGLE, students are given AABC. They should construct the altitude of each vertex of the tri-
angle. Students should realize that they are concurrent. Explain to students that the point of concurrency
is called the orthocenter of the triangle.

Students should discover a few facts about the orthocenter. If a triangle is an acute triangle, then the
orthocenter is inside of the triangle. If a triangle is a right triangle, then the orthocenter is on a side of
the triangle. If a triangle is an obtuse triangle, then the orthocenter is outside of the triangle.

/ .-■■* T-vti..
-■ s i, --s

_-- T

www.ckl2.org 34

Problem 3 — Exploring the Altitude of an Equilateral Triangle

Students will need to find the distance from point P to the 3 sides of the triangle and the altitude BD in
the file EQUILATE. The Distance & Length tool is found in the GRAPH/F5 menu (press GRAPH
and select Measure > D.&Tength). Once students make their calculation, they can move it to the right
of the appropriate label (EP, for instance). Students will then need to calculate the sum of EP + FP + GP
using the Calculate tool (press GRAPH and select Calculate). Students will need to move their cursor
to EP until it is underlined, then press ENTER, then press +, then move their cursor to FP until it
is underlined, then press ENTER. A measurement that you can place anywhere on the screen is given.
Place the measurement by pressing ENTER. Repeat this process for EP + FP that was just created and
GP to get the sum EP + FP + GP. Students should discover that the sum of the distances from all three
sides to point P and the length of the altitude BD are equal. Make sure that students see that this only
works for an equilateral triangle.

Problem 4 — Exploring the Orthocenter of a Medial Triangle

In the file MEDIAL2, students are given a triangle, its medial triangle, and the orthocenter of the medial
triangle. Students are to discover which point of the original AABC is the orthocenter of the medial triangle.
Choices should include the centroid, circumcenter, incenter, and orthocenter.

Students should discover that the orthocenter of the medial triangle is the circumcenter of the original
triangle.

1 ,,,*

"'■-..

if-"

'<£&<*>

Solutions

1. An altitude of a triangle is a perpendicular segment from a vertex to the line containing the side opposite
that vertex.

2.

A C

3. a. inside

b. outside

c. a side of

4. They are concurrent

5. Right Triangle

6. Acute Triangle

35

www.ckl2.org

7. Obtuse Triangle

Table 6.2:

Position

1 st position

2 nd position

3 rd position

A th position

BD

EP + FP + GP

4.9
4.9

5.4
5.4

5.4
5.4

5.4
5.4

9. They are equal

10. equal to the length of the altitude

11. circumcenter

www.ckl2.org 36

Chapter 7

7.1 Properties of Parallelograms

This activity is intended to supplement Geometry, Chapter 6, Lesson 2.

ID: 11932

Time Required: 15 minutes

Activity Overview

Students will explore the various properties of parallelograms. As an extension, students can also explore
necessary and sufficient conditions that guarantee that a quadrilateral is a parallelogram.

• Inductive Reasoning

• Parallelograms

Teacher Preparation and Notes

• This activity was written to be explored with the Cabri Jr. app on the TI-84-

• Before beginning this activity, make sure that all students have the Cabri Jr. application, and the
Cabri Jr. files PAR1.8xv, PAR2.8xv, PAR3.8xv loaded on their TI-84 calculators.

• In Cabri Jr., to grab a point hover the cursor over the point and press ALPHA. To release press
ALPHA or ENTER. To move a point after grabbing it, use the arrow keys.

Software/Detail? id=258#.

US/Activities/Detail?id=11932 and select PAR, PAR2, and PAR3.

Associated Materials

• Student Worksheet: Properties of Parallelograms http://www.ckl2.org/flexr/chapter/9691

• Cabri Jr. Application

. PAR.8xv, PAR2.8xv, and PAR3.8xv

37 www.ckl2.org

Problem 1 — Properties of Parallelograms

Students will begin this activity by looking at properties of parallelograms. They will discover that opposite
sides are congruent, opposite angles are congruent, and that consecutive angles are supplementary.

As an extension, students can prove each of these using parallel lines and transversals. Students will need
to know the properties of alternate interior angles, same-side interior angles, and corresponding angles.

Problem 2 — Diagonals of Parallelograms

In Problem 2, students are asked to investigate the diagonals of a parallelogram. Students should discover
that the diagonals of a parallelogram bisect each other. This particular wording may be hard for students
discover independently.

Problem 3 — Extension: Proving Parallelograms

In this problem, students can explore various properties and see if they guarantee that a quadrilateral is a
parallelogram.

Students should know that the following prove that a quadrilateral is a parallelogram:

1. both pairs of opposite sides are congruent

2. both pairs of opposite angles are congruent

3. both pairs of opposite sides are parallel

4. one pair of opposite sides is both parallel and congruent

5. the diagonals bisect each other

Problem 4 — Extension: Extending the Properties

For this problem, students will create any quadrilateral and name it GEAR. Next, students will find the
midpoint of each side and connect the midpoints to form a quadrilateral. Students will use the properties
of a parallelogram to see that a parallelogram is always created.

Solutions

1. A quadrilateral with both pairs of opposite sides parallel.

Table 7.1:

Position

QU

UA

DQ

1

4.50

2.44

4.50

2.44

2

4.50

2.94

4.50

2.94

3

4.90

2.94

4.90

2.94

4

5.50

2.94

5.50

2.94

3. The opposite sides are congruent.

www.ckl2.org 38

Table 7.2:

Position

LQ

L13

LA

ID

1

100

80

100

80

2

106

74

106

74

3

124

56

124

56

4

77

103

77

103

5. The consecutive angles are supplementary.

6. The opposite angles are congruent.

Table 7.3:

Position

QR

AR

DR

RU

1

2.84

2.84

3.29

3.29

2

2.94

2.94

3.45

3.45

3

3.06

3.06

3.58

3.58

4

3.19

3.19

3.71

3.71

8. The diagonals bisect each other.

9. Yes

10. No

11. Yes

12. Parallelogram

13. Sample: Both pairs of opposite sides are parallel.

7.2 Properties of Trapezoids and Isosceles Trape-
zoids

This activity is intended to supplement Geometry, Chapter 6, Lesson 5.

ID: 9084

Time required: 25 minutes

• Prove and apply theorems about the properties of rhombuses, kites and trapezoids.

Activity Overview

A trapezoid is a quadrilateral where one pair of sides is parallel while the other two sides are not. In
an isosceles trapezoid the non-parallel sides are congruent. In this activity we will attempt to create an

39

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isosceles trapezoid from an ordinary trapezoid, then approach the problem in a different manner and finally,
examine the properties of trapezoids.

Teacher Preparation

This activity is designed to be used in a middle-school or high-school geometry classroom.

• For this activity, students should know the definitions of a trapezoid and isosceles trapezoid. If they
do not already know these terms, you can define them as they appear in the lesson, but allow extra
time to do so.

Software/Detail? id=258#.

Associated Materials

• Student Worksheet: Properties of Trapezoids and Isosceles Trapezoids http : //www . ckl2 . org/f lexr/
chapter/9691, scroll down to the second activity.

• Cabri Jr. Application

Classroom Management

• This activity is intended to be mainly teacher-led, with breaks for individual student work. Use the
following pages to present the material to the class and encourage discussion. Students will follow
along using their graphing calculators.

• See the student worksheet on how to complete this activity.

7.3 Properties of Rhombi, Kites and Trapezoids

ID: 12093

Time Required: 45 minutes

Activity Overview

In this activity, students will discover properties of the diagonals of rhombi and kites and properties of
angles in rhombi, kites, and trapezoids.

• Rhombi

• Kites

• Trapezoids

Teacher Preparation and Notes

• This activity was written to be explored with the Cabri Jr. app on the TI-84-

Software/Detail?id=258#.

KING 3, and TRAP.

www.ckl2.org 40

Associated Materials

• Student Worksheet: Properties of Rhombi, Kites and Trapezoids http://www.ckl2.org/flexr/
chapter/9691, scroll down to the third activity.

• Cabri Jr. Application

. KING1.8xv, KING2.8xv, KING3.8xv
. TRAP.8xv

Problem 1 — Properties of Rhombi

Students will begin this activity by looking at angle properties of rhombi. They are asked several questions
about the angles and diagonals of a rhombus.

An extension of this exercise would be to prove each of these using parallel lines and transversals. Students
will need to know the properties of alternate interior angles, same-side interior angles, and corresponding
angles.

Problem 2 — Properties of Kites

In Problem 2, students will be asked to begin looking at angle properties of kites. They are asked several
questions about the angles and diagonals of kites.

Problem 3 — Properties of Trapezoids

In this problem, they will explore angle properties of trapezoids. Students are given trapezoid TRAP and
the measure of angles T, R, A, and P. Students will move point R to four different positions and collect
the measures of angles T, R, A, and P onto their accompanying worksheet.

An extension of this exercise would be to prove leg angles are supplementary using parallel lines and
transversals. Students will need to know the properties of alternate interior angles, same-side interior
angles, and corresponding angles.

Solutions

Table 7.4:

Position

LR

LE

lA

LD

1
2
3
4

76
51
95
64

104
129

85
116

76
51
95
64

104
128
85
116

2. supplementary

3. congruent

41

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4. right angles

5. bisect

Table 7.5:

Position

IK

Ll

IN

IG

1
2
3
4

69

71
61
37

130
116
123

142

32
58
53
38

130
116
123

142

7. One pair of opposite angles is congruent and one pair of opposite angles is not congruent.

8. right angles

9. One pair of opposite angles is bisected and one set of opposite angle is not. The pair that is not
congruent is the one getting bisected.

Table 7.6:

Position

T

1

108

2

126

3

39

4

57

R

72
54
141
123

54
45
24
37

126
135
156
143

11. Leg angles are supplementary.

www.ckl2.org

42

Chapter 8

TE Similarity - TI

8.1 Constructing Similar Triangles

This activity is intended to supplement Geometry, Chapter 7, Lesson 4-

ID: 8179

Time required: 30 minutes

Activity Overview

In this activity, students will investigate three different methods of constructing similar triangles. They
will use the Dilation tool with the dilation point inside and outside of the triangle to investigate different
relationships. Also, students will use the Parallel tool to construct two similar triangles from one triangle.

Topic: Ratio, Proportion & Similarity

• Use inductive reasoning to classify each set of conditions as necessary and/or sufficient for similarity:

a) the side lengths of one triangle are equal to the corresponding side lengths of another triangle.

b) two triangles of one triangle are congruent to two angles of another triangle.
Teacher Preparation

• This activity is designed to be used in a high school geometry classroom.

• Before starting this activity, students should be familiar with the term dilation.

• This activity is intended to be mainly teacher-led, with breaks for individual student work. Use the
following pages to present the material to the class and encourage discussion. Students will follow
along using their calculators.

Software/Detail? id=258#.

Associated Materials

• Student Worksheet: Constructing Similar Triangles http://www.ckl2.org/flexr/chapter/9692

• Cabri Jr. Application

43 www.ckl2.org

Students need to press APPS and select CabriJr to start the application. When they open a new
document, they need to make sure that the axes are hidden. If the axes are displayed, press GRAPH >
Hide/Show > Axes.

Explain to students that similar triangles are those that have the same shape but not necessarily the same
size.

Congruent triangles are a special type of similar triangle where corresponding sides are congruent. In
similar triangles, corresponding angles are congruent but corresponding sides are proportional. In this
activity, students will look at three methods of constructing similar triangles and will test these properties
using dilations or stretches.

In order to examine all of the sides and angles, students should work in groups of three. Have one student
(Student A on the worksheet) in each group construct the first triangle and save it as "SIMTRI." This is
saved in the TI-84 Plus family as an APPVAR. Student A needs to transfer the APPVAR to the other
two students in the group.

Problem 1 — Similar Triangles using Dilation

Student A will use the Triangle tool to construct a triangle of any shape or size. Then they need to use
the Alph-Num tool to label the vertices P, Q, and R.

Student A should measure angle P and side PQ, Student B should measure angle Q and side QR, Student
C should measure angle R and side PR.

Note: To increase the number of digits for the length of the side, hover the cursor over the measurement
and press +.

With the Point tool, students on their own calculators will construct a point C in the center of the triangle.
They will then use the Alph-Num tool to place the number 2 at the top of the screen.

Explain to students that the point C will be the center of the dilation and the number 2 will be the scale
factor.

-i-il F-Q E.iH

A,

With the Dilation tool selected, students need to move the cursor to point C, the center of the dilation,
and press ENTER. They should notice that the shape of the cursor changes, which is to indicate that
everything will be stretched away from this point. Then students move the cursor to the perimeter of the
triangle and press ENTER. Finally, move to the scale factor, 2, and press ENTER.

Students will see that a new, larger triangle will appear outside of APQR. Have a prelimenary discussion
with students about whether they think the new triangle is similar to APQR and how they can confrim
their hypothesis.

www.ckl2.org 44

A\

Instruct students to label this triangle as XYZ so that X corresponds to P, Y to Q and Z to R. Each person
in the group should select and measure their appropriate angle and side in the new triangle. Students
should answer Questions 1 and 2 on the worksheet comparing the two angles and two sides.

Explain to students that XY = 2 PQ indicates that the sides have the ratio 2:1. If all three sides display
the same result, then the sides are said to be proportional.

To answer Questions 3 and 4 on the worksheet, students will observe the changes in the triangles as they
drag a point in the original triangle and the point of dilation.

Problem 2 — Different Scale Factors

Students will continue using the same file. To change the scale factor, students must select the Alph-Num
tool, move the cursor to the scale factor of 2 and press ENTER. Then they should delete 2 by pressing
DEL, press ALPHA to change the character to a number, and enter 3. Students can now answer Question
5 on the worksheet.

Now students will investigate another scale factor by changing it from 3 to 0.5. They should then answer
Questions 6 and 7 on the worksheet, summarizing their observations of the triangles and their measure-
ments.

For the last investigation in this problem, students will look at the effect of a dilation when the center
point C is outside of the pre-image triangle and a negative scale factor is used. Students will need to move
APQR by moving the cursor to one side and when all sides of the triangle flash, press ALPHA and then
use the arrow keys.

Problem 3 — Similar Triangles with a Parallel Line

Finally, students will look at a completely different method of constructing similar triangles. All students
will need to open a new file. It is the teacher's decision to have them save the file. Student A should
construct a triangle PQR and transfer it to the others in their group. They will all measure their same side
and angle as before.

,.' ■■■,

P fiH*

PQ T-.72

_-^ F:

„/-

All students need to construct a point on PQ using the Point on tool and label it S using the Alph-Num
tool. Using this point, they construct a line that is parallel to QR by selecting the Parallel tool, and then
choosing the point and side.

45 www.ckl2.org

With the Intersection tool, students need to find the intersection of side PR and the parallel line, and
then use the Alph-Num tool to label it as T. The Hide/Show > Object tool enables students to hide
the parallel line, and then construct segment S T with the Segment tool. If students can prove that all of
the angles are congruent, they have completed their first step in proving that the two triangles are similar.

With the Measure > D. & Length tool, students need to measure segments of the corresponding sides.
In the screen to the right all three pairs of sides have been measured.

To complete the proof, students will need to confirm that all of the sides are proportional. Using the Calcu-
late tool, they can select a side measurement, press the operation key (-r) and then select the corresponding
side measurement.

If all three ratios are equivalent, then the sides are proportional and the two triangles are similar.

8.2 Side-Splitter Theorem

This activity is intended to supplement Geometry, Chapter 7, Lesson 5.

ID: 12318

Time Required: 15 minutes

Activity Overview

In this activity, students will explore the side-splitter theorem.
Topic: Ratio, Proportion &; Similarity

• Side-Splitter Theorem
Teacher Preparation and Notes

• This activity was written to be explored with the Cabri Jr. app on the TI-84-

Software/Detail? id=258#.

US/ 'Activities /Detail? , id=12318 and select SIDESP1, SIDESP2, and SIDESP3.

Associated Materials

• Student Worksheet: Side-Splitter Theorem http : //www . ckl2 . org/f lexr/chapter/9692, scroll down
to the second activity.

• Cabri Jr. Application

. SIDESP1.8xv, SIDESP2.8xv, and SIDESP3.8xv

Problem 1 — Side-Splitter Theorem

Students will begin this activity by looking at the side-splitter theorem. Students are given a triangle with
a segment parallel to one side. They will discover that if a line is parallel to one side of a triangle and
intersects the other two sides, then it divides those sides proportionally.

www.ckl2.org 46

Students will be asked to collect data by moving point A and point D. Students are asked several questions
about the relationships in the triangle.

.PA

DC i

V

~T

x

HS H.S
Sft i.h

Problem 2 — Application of the Side-Splitter Theorem

In Problem 2, students will be asked to apply the side-splitter theorem to several homework problems.

Problem 3 — Extension of the Side-Splitter Theorem

In Problem 3, students will discover the corollary to the Side-Splitter Theorem: If a line is parallel to one
side of a triangle and intersects the other two sides, then it divides those sides proportionally.

Students are asked several questions about the corollary to the side-splitter theorem.

Solutions

Table 8.1:

Position

DC

AS

SR

DC

AS
SR

1

4.1

1.8

4.6

2.0

2.27

2.30

2

3

2.8

3.4

3.2

1.07

1.06

3

2.6

2.4

3.2

3.0

1.08

1.07

4

3.3

3.1

2.5

2.3

1.06

1.08

2. The ratios of the side lengths are equal.

°- DC ~ SR

4. The ratio remains the same.

5. The ratio changes when moving point D.

6. When you move the parallel line, you are changing the proportion between the upper and lower segments.
When you move the point, the segments may get longer or shorter, but the proportion stays the same.

7. 12

8. 16.8

47

www.ckl2.org

Table 8.2:

Position

RN

NO

EA

AS

RN

NO

A5

1

2.1

1.5

2.4

1.8

1.4

1.4

2

2.1

1.3

2.7

1.7

1.6

1.6

3

1.9

1.5

2.4

2.0

1.2

1.2

4

2.7

0.7

3.5

0.9

4

4

10. The ratios are equal.

11. ^ and fr are congruent

8.3 Perspective Drawing

TTiz's activity can be used to supplement the "Know What?" from Chapter 7, Lesson 6. 1

The student worksheets for this activity are at the end of Chapter 12 because this activity can also be used
at the end of the year. The link is below.

ID: 10033

Time required: 60 minutes

Activity Overview

In this activity, students draw figures in one- and two-point perspective and compare and contrast the two
types of drawings. They then create an isometric drawing and compare it to their drawings in perspective.

Topic: 3— Dimensional Geometry

• Construct 3- dimensional prisms and pyramids.

• Record the number of faces, edges, and vertices of prisms.

Teacher Preparation and Notes

• Perspective drawings can be taught at any time in a geometry curriculum, but are most appropriate
after lessons on parallel and perpendicular lines, three-dimensional figures, and symmetry.

• Throughout the activity, students use many drawing and construction tools, such as the Segment,
Parallel, and Perpendicular tools. In this document, the first use of a tool is by name and accom-
panied by its location within the menu structure. For subsequent uses, the tool may be mentioned by
name or its function, and the menu location is omitted.

• This activity is designed to have students explore individually or in pairs.

Software/Detail?id=258#.

US/Activities/Detail?id=10033 and select RECPRISM, TRIPRISM, TWOPERSP, ISO-
DRAW.

Associated Materials

www.ckl2.org 48

• Student Worksheet: Perspective Drawing http://www.ckl2.org/flexr/chapter/9697, scroll down
to the third activity.

• Cabri Jr. Application

. RECPRISM.8xv, TRIPRISM.8xv, TW0PERSP.8xv, and IS0DRAW.8xv

Problem 1 — One-point perspective

Students will open the Cabri Jr. file RECPRISM and find a vaishing point A along the horizon and a
rectangle BCDE.

They are to use the Segment tool from the F2 menu to draw AB and AC. These two segments (and any
other segment that joins a vertex of the prism to the vanishing point) are hereafter referred to as vanishing
segments.

Next, they will create FG such that F is between A and B, G is between A and C, and FG X BC (Refer to
the diagram at the right).

To do this, they will need to do the following:

• Place a point F on AB. (F2 > Point > Point On)

Construct a line through F parallel to BC.
(F3 > Parallel)

• Plot point G at the intersection of the parallel line and AC. (F2 > Point > Intersection)

• Hide the line. (F5 > Hide/Show > Object)

• Draw FG.

Students can now hide the vanishing segments, AB and AC, and draw BF and CG. The front and top faces
of the rectangular prism have been constructed.

Ask students if they are satisfied that this drawing completely represents a rectangular prism. Have them
drag point A (press APLHA to grab the point) to either the far left or far right and ask students what is
wrong with the drawing. (The rectangular prism does not have a face on either side.)

To complete the prism, students next need to construct AD and AE. Then, using the Perpendicular tool
(F3 > Perp.), construct two lines perpendicular to FG — one through F and the other through G. The
vertical "back" edges GH and FJ may then be created by plotting the intersection points, hiding the lines,
and constructing the segments as needed.

To show all six edges, students should draw a segment connecting H and /. Once all of the vanishing
segments are hidden, students will find they need to also construct segments DH and EJ.

H

-Q-

& n-'"'"Ji J yi H

Depending on the topics you have recently covered in class, you can now have students identify parallel,
intersecting, and skew lines; count faces, vertices, and edges; or discuss the similarity of rectangles BCDE
and FGHJ.

49 www.ckl2.org

When students have finished exploring their prisms, they may wish to hide or dash the edges that would
not be visible if the figure were not transparent. The appearance of the edges may be altered using the
Display tool (F5 > Display).

Students are to open TRIPRISM and create a triangular prism in one-point perspective on their own.
Be sure to circulate around the room and assist students as needed.

Allow students a few minutes to explore the figure by dragging point A along the horizon or dragging an
edge. Vertices B, C, and D can also be dragged on this figure.

Problem 2 — Two-point perspective

The Cabri Jr. file TWOPERSP contains two vanishing points, A and B, signifying this rectangular prism
will be drawn in two-point perspective. The file also contains a vertical line segment, CD, which will be
the front edge of the prism.

CD is constructed so that dragging point C changes the segment's length and dragging point D translates
the segment.

Next, students will then use the Segment tool to draw the vanishing segments AC, AD, BC, and BD.
Then they should use the Point On, Parallel, Intersection, Hide/Show, and Segment tools (as they
did in Problem 1) to construct the vertical edges, EF and GH, of the left and right faces.

The top and bottom edges of those faces, CE, CG, DF, and DH may be drawn as well.

Students should now hide the four vanishing segments, and then draw four more: from E and F to B and
from G and H to A.

The intersection of the upper two vanishing segments and that of the lower two vanishing segments should
be plotted and a segment drawn between them to form the last vertical edge, JK.

I

~w

The vanishing segments may then be hidden so that students can draw the remaining edges of the prism.

As before, allow students some time to drag edges, vertices, and vanishing points, observing how the prism
changes as they do so. (Dragging CD is very interesting!) Again, students can either hide or dash any
"hidden edges."

Problem 3 — An isometric drawing

Using ISODRAW, students will construct an isometric drawing of a rectangular prism. The three seg-
ments shown represent the three "front" edges of the prism, and each of the angles the segments form
measures 120°. Students should first use the Parallel, Intersection, Hide/Show, and Segment tools
to construct AE, CF, DE, and DF, as shown in the diagram to the right.

They may then use the same sequence of tools once more to construct the remaining edges: AG, CG, EH,
~FH, and GH.

www.ckl2.org 50

r

After completing the isometric drawing, students should compare it to the perspective drawings from
Problems 1 and 2. As a similarity, students may identify that right angles "in real-life" do not actually
measure 90° in the drawings. As a difference, all parallel lines "in real-life" are parallel in an isometric
drawing, whereas in perspective drawings, some parallel lines actually "meet" at the vanishing point.

51

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Chapter 9

TE Right Triangle
Trigonometry - TI

9.1 The Pythagorean Theorem

This activity is intended to supplement Geometry, Chapter 8, Lesson 1.

ID: 9532

Time required: 60 minutes

Activity Overview

In this activity, students will use the Cabri Jr. application to construct figures that prove the Pythagorean
Theorem in two different ways.

Topic: Right Triangles & Trigonometric Ratios

• Construct and measure the side lengths of several right triangles and conjecture a relationship between
the areas of the squares drawn on each side.

• Prove and apply the Pythagorean Theorem.

Teacher Preparation

• This activity is designed to be used in a high school or middle school geometry classroom.

• This activity is designed to be student- centered with the teacher acting as a facilitator while students
work cooperatively. Use the following pages as a framework as to how the activity will progress. Feel
free to print out the following pages for your students.

• The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse
equals the sum of the squares of the lengths of the legs. This can be expressed as c 2 = a 2 + b 2 where
c is the length of the hypotenuse.

• Depending on student skill level, you may wish to download the constructed figures to student cal-
culators. If the files are downloaded, skip the construction steps for each problem and begin each at
Step 10.

• Note: Measurements can display 0, 1, or 2 decimal digits. If digits are displayed, the value shown
will round from the actual value. To change the number of digits displayed:

1. Move the cursor over the value so it is highlighted.

www.ckl2.org 52

2. Press + to display additional decimal digits or - to hide digits.

Software/Detail?id=258#.

US/Activities/Detail?id=9532 and select PYTHAG1 and PYTHAG2.

Associated Materials

• Student Worksheet: The Pythagorean Theorem http://www.ckl2.org/flexr/chapter/9693

• Cabri Jr. Application

. PYTHAG1.8xv and PYTHAG2.8xv

Problem 1 — Squares on Sides Proof

The Pythagorean Theorem states that, the square of the length of the hypotenuse of a right triangle is
equal to the sum of the squares of the legs. In this activity, you will construct a right triangle and verify
the Pythagorean Theorem by constructing squares on each side and comparing the sum of the area of the
two smaller squares to the area of the square of the third side.

Step 1: Open a new Cabri Jr. file.

Construct a segment using the Segment tool.

Select the Alph-Num tool to label the endpoints B and C as shown.

Step 2: Construct a line through C that is perpendicular to BC using the Perp. tool.

Step 3: Construct a point on the perpendicular line and label it A.

Hide the perpendicular line with the Hide/Show tool and construct line segments AC and AB.

For the time being, keep the sides of the triangle fairly small so that squares can be constructed on the
sides.

Step 4: In the lower left corner, use the Alph-Num tool to place the number 90 on the screen. This will
be the angle of rotation.

Note: Press ALPHA to access numerical characters. A small "1" will appear in the tool icon in the
upper left corner of the screen.

Step 5: Use the Rotation tool to rotate point C about point B through an angle of 90°.

• Press ENTER on point B as the center of rotation.

• Press ENTER on the angle of rotation (the number 90).

• Press ENTER on point C, the object to be rotated.

Notice that the number now has a degree symbol associated with it and that the point has been rotated
in the counter-clockwise direction.

b^

90* o fc

53 www.ckl2.org

Step 6: What we want to do next is to rotate point B about point C through an angle of 90° in the
clockwise direction. To do this we will need an angle of -90. Place this number on the screen.

Using the value of -90, rotate point B about point C through an angle of -90°.

Step 7: You should now have two points below the line segment BC. Use the Quad, tool to construct a
quadrilateral using points B, C and the two points constructed in Steps 5 and 6.

Answer Question 1 on the worksheet.

Step 8: In a similar fashion, rotate point C about point A through an angle of -90° and rotate point A
about point C through an angle of 90°. This will allow us to construct a second square.

Use the Quad, tool again to construct the square on side AC.

Step 9: Finally, rotate point B about point A through an angle of 90° and rotate point A about point B
through an angle of -90°.

Then construct a third square on hypotenuse AB.

n

c

-90*
Big-

step 10: Start with this step if you are using the pre- constructed file "PYTHAG1".

Select the Measure > Area tool and measure the area of the three squares.

Step 11: Using the Calculate tool, press ENTER on the measurements of the two smaller squares and
then press the + key. Place the sum off to the side of the screen.

How does this sum compare to the square of the hypotenuse? Record your observations in the table for
Question 2 on the worksheet.

Step 12: To test your construction, drag points A, B and/or C to a new location on the screen.

Answer Question 3 on the worksheet.

Problem 2 — Inside a Square Proof

In this problem, we are going to look at a proof of the Pythagorean Theorem. We hope to prove the
statement that, if z is the length of the hypotenuse of a right triangle and x and y are the lengths of the
legs of the right triangle, then z 2 = x 2 + y 2 .

Step 1: Construct a line segment AB.

Use the Alph-Num tool to place the value 90 on your screen.

fin oE:

90 fc

www.ckl2.org 54

Step 2: Access the Rotation tool and press ENTER on point A as the center of the rotation, then on
90 as the angle of rotation and finally on line segment AB as the object to be rotated.

Label the new point D.

Step 3: Continue by rotating line segment AD about point D through an angle of 90°.

Label the new point C.

Step 4: Complete the square by constructing line segment BC.

Step 5: Using the Point on tool, add point E on AB as shown and overlay a line segment BE.

Dp

qiv:

Step 6: Select the Compass tool to construct circles with radius equal to the length of BE.

• Press ENTER on BE. A dashed circle will appear and follow the pointer.

• Press ENTER on point A. The compass circle is anchored at center A.

Create a point of intersection of this circle with AD. Label this point F.
Hide the compass circle.

['?

<?'-

{*H-

Q '■

Step 7: Use the Compass tool again to construct circles with centers at C and D and radius = BE.

Create points of intersection of these circles with DC and BC. Label these points G and H.

Hide the compass circles.

Drag point E to confirm that F, G, and H all move as E moves.

Step 8: Construct the quadrilateral EFGH. Can you prove that this quadrilateral is a square?

n<i--

Step 9: Use the Alph-Num tool to place the labels x, y, and z on the figure as shown.
BE is labeled x, therefore AF = DG = CH = x.

55

www.ckl2.org

EA is labeled y, therefore FD = GC = HB = y.

Since ABCD is a square, each of the angles at A, B, C and D are 90°, so we have four congruent triangles,
namely AEFA, aFGD, aGHC and AHEB.

qfi-:

V eK

Step 10: Start with this step if you are using the pre- constructed file "PYTHAG2".

Let's examine the algebra in this situation.

ABCD is a square with sides of length (x + y) .

The area of the square ABCD is (x + y) 2 = x 2 + 2xy + y 2 .

Each of the triangles AEFA, aFGD, aGHC and AHEB is a right triangle with height x and base y. So, the
area of each triangle is -^xy. The sum of the areas of the four triangles is 4 • ^xy = 2xy.

EFGH is a square with sides of length z- So the area of EFGH is z 2 .

Looking at the areas in the diagram we can conclude that:

ABCD = AEFA + AFGD + AGHC + AHEB + £FG#

On the worksheet, substitute the area expressions (with variables x, y, and z) into the equation above and
simplify.

Step 11: Let's look at this numerically as well to confirm what we just proved algebraically. Measure BE,
~HB and EH.

Note: Measure HB and EH by pressing ENTER on each endpoint, since these do not have separate
segments constructed.

Use the Calculate tool to find the squares of these lengths.

Record your observations in the table for Question 6 on the worksheet.

Step 12: Find the sum of the squares of the lengths of segments BE and HB.

In the right triangle HBE, is BE 2 + HB 2 = EH 2 ?

Drag point E to ensure that the relationship holds for other locations of the points E, F, G, and H.

What would happen if you dragged one of points A or fi? Would the relationship still hold?

Answers Questions 7 and 8 on the worksheet.

9.2 Investigating Special Triangles

This activity is intended to supplement Geometry, Chapter 8, Lesson 4-

ID: 7896

Time required: 45 minutes

www.ckl2.org 56

Activity Overview

In this activity, students will investigate the properties of an isosceles triangle. Then students will construct
a 30° - 60° - 90° triangle and a 45° - 45° - 90° 'triangle to explore the ratios of the lengths of the sides.

Topic: Right Triangles & Trigonometric Ratios

• Calculate the trigonometric ratios for 45° - 45° - 90°, 60° - 60° - 60° and 30° - 60° - 90° triangles.

Teacher Preparation and Notes

This activity is designed to be used in a high school or middle school geometry classroom.

• If needed, review or introduce the term median of a triangle. Any median of an equilateral triangle
is also an altitude, angle bisector, and perpendicular bisector.

• This activity is designed to be student- centered with the teacher acting as a facilitator while students
work cooperatively.

• The worksheet guides students through the main ideas of the activity and provides a place for students
to record their work. You may wish to have the class record their answers on separate sheets of paper,
or just use the questions posed to engage a class discussion.

Software/Detail? id=258#.

US/Activities/Detail?id=7896 and select EQUIL and ISOSC.

Associated Materials

• Student Worksheet: Investigating Special Right Triangles http://www.ckl2.org/flexr/chapter/
9693, scroll down to the second activity.

• Cabri Jr. Application

. EQUIL.8xv and ISOSC.8xv

Problem 1 — Investigation of Triangles

First, turn on your TI-84 and press APPS. Arrow down until you see Cabri Jr and press ENTER. Open
the file ISOSC. This file has a triangle with an isosceles triangle with AB = AC.

Using the Perpendicular tool (ZOOM > Perp.), construct a perpendicular from point A to side BC.
Label the point of intersection of this line with BC as D. To name the point, they need to select the
Alph-Num tool (GRAPH > Alph-Num), select the point, and press x _1 ENTER for the letter D.

Construct line segments BD and CD (WINDOW > Segment) and then measure the segments (GRAPH
> Measure > D. & Length).

57 www.ckl2.org

Would you have expected these segments to be equal in length?

Drag point C to see the effect on the lengths of the line segments. It appears that the perpendicular from
the vertex always bisects the opposite side. Measure the angles BAD and CAD.

Will they always be equal?

Problem 2 — Investigation of 30-60-90 Triangles

Open the file EQUIL. Note that all three angles are 60° angles.

Construct the perpendicular from A to side BC. Label the point of intersection as D, like in Problem 1.

From the construction above, we know that D bisects BC and that mlBAD = 30°.

Construct segment BD. We now have triangle BAD where mlD = 90°, mlB = 60° and mlA = 30°. We also
have triangle ACD where mlA = 30°, mlC = 60° and mlD = 90°.

This completes the construction of two 30° - 60° - 90° triangles. We will work only with the triangle BAD.

You may choose to have the students hide the segments AC and CD. To do this, construct
segments BD and AB on top of the larger triangle. Then hide the original triangle. Keep the
point C. We will need that point later to resize the triangle.

Measure the three sides of the triangle.

Press GRAPH and select the Calculate tool. Click on the length of BD, then on the length of AB.
Press the -f key. Students will see the result 0.5. Move it to the upper corner. Repeat this step to
find the ratio of AD : AB and AD : BD. These ratios will become important when you start working with
trigonometry.

Drag point C to another location.

H.90/

£.15 D

i'i 50
0.B7

1.73

4.2H |t

What do you notice about the three ratios?

Problem 3 — Investigation of 45-45-90 Triangles

Press the Y = button and select New to open a new document.

To begin the construction of the 45° - 45° - 90° triangle, construct line segment AB and a perpendicular
to AB at A.

Use the compass tool with center A and radius AB. The circle will intersect the perpendicular line at C.
www.ckl2.org 58

Hide the circle and construct segments AC and BC.

Can you explain why AB = AC and why angle ACB = angle ABC?

Why are these two angles 45° each?

Students should notice that the two angles must be equal, and angle A is 90°. Therefore,
because the sum of the angles in a triangle is 180°, the two angles must be 45° each.

Measure the sides of the triangle. This verifies that AB = AC.

Use the CALCULATE tool to find the ratio of AC : BC and AC : AB. Once again, these ratios will be
important when you study trigonometry.

:-:.bm

^

0.?i
i.Od

\

SiB"

J.BO

Drag point B and observe what happens to the sides and ratios.

Why do the ratios remain constant while the sides change?

Students should notice that AC and AB are equal, so the ratios will always remain the same.

9.3 Ratios of Right Triangles

This activity is intended to supplement Geometry, Chapter 8, Lesson 5.

ID: 11576

Time Required: 45 minutes

Activity Overview

In this activity, students will explore the ratios of right triangles. Students will discover that they can find
the measure of the angles of a right triangle given the length of any two sides.

Topic: Right Triangles & Trigonometric Ratios

• Sine

• Cosine

• Tangent

Teacher Preparation and Notes

59 www.ckl2.org

• This activity was written to be explored on the TI-84 with the Cabri Jr. and Learning Check appli-
cations.

• Before beginning this activity, make sure that all students have the Cabri Jr. applications. Also,
make sure students have or know the trigonometric definitions.

Software/Detail? id=258#.

US/ 'Activities/Detail? id=11576 and select TRIG.

Associated Materials

Student Worksheet: Ratios of Right Triangles http://www.ckl2.org/flexr/chapter/9693, scroll
down to the third activity.
Cabri Jr. Application
TRIG.8xv

Problem 1 — Exploring Right Triangle Trigonometry

You may need to allow students to use a textbook (or other resource) to hnd the definitions of sine, cosine,
and tangent.

Students are asked to give the ratio of several triangles on their handheld or their accompanying worksheet.

Problem 2 — Exploring the Sine Ratio of a Right Triangle

For this problem, students will investigate the sine ratio of two sides of a triangle. Students should start
the Cabri Jr. app and open the file Trig.8xv.

Students will collect data on their worksheets by moving point B. They will do this for four different
positions of the point.

ME \$.2
EC £.2

ML 1.7

Htfc:

BtYAB O.HE
hlYAB ij,9i
BC/hC if.H?

Students will discover that the ratio of BC to AB remains constant, no matter how large the triangle is;
Therefore, students will be able to use the inverse of sine to find the measure of the angles in AABC.

Students will need to answer several questions on their handhelds or their accompanying worksheets.

Problem 3 — Exploring the Cosine Ratio of a Right Triangle

Students will repeat the exploration in Problem 2, but with the cosine ratio.
www.ckl2.org 60

Problem 4 — Applying the Sine, Cosine, and Tangent Ratio of a
Right Triangle

In Problem 4, students are asked to apply what they have learned about how to find the measure of an
angle of a right triangle given two sides of the triangle.

£in-K2/5)

.4115168461
sin-K2/5>

23.57817843
cos-K2/"5)

66.42182152

Solutions

1. For right triangle ABC, the sine of an angle is the ratio of the length of the opposite side to the length
of the hypotenuse.

2. For right triangle ABC, the cosine of an angle is the ratio of the length of the adjacent side to the length
of the hypotenuse.

3. For right triangle ABC, the tangent of an angle is the ratio of the length of the opposite side to the

4-1
M

M

M
9. |

Table 9.1:

Position

BC

AB

BC
AB

sin

■1 BC

AB

1
2

3

4

2.4376781463393
3.0769811671077
3.6665092204124
4.3154341679767

6.2006451991814
7.8268201774092
9.3263841370716
10.977034107736

0.39313298342907
0.39313298342907
0.39313298342906
0.39313298342905

23.149583787224
23.149583787224
23.149583787223
23.149583787222

11. The ratio does not change.

12. No, the angle does not change.

13. 23.1496

14. 66.8504

61

www.ckl2.org

Table 9.2:

Position AC AB % cos" 1 %

1 7.0816099136391 8.8549125969341 0.7997379800328 36.894911430193

2 8.0238624186986 10.033114118664 0.79973798003277 36.894911430196

3 9.0235078139592 11.283080257848 0.79973798003279 36.894911430194

4 3.7704816328074 4.7146462053143 0.79973798003281 36.894911430192

16. 36.8949

17. 53.1051

18. A = tan" 1 f§

19. A = 23.57,5 = 66.42

20. A = 21.8,5 = 68.2

21. A = 23.96,5 = 66.04

22. A = 53.13,5 = 36.87

23. A = 15.07,5 = 74.93

24. A = 42.83,5 = 47.17

25. A = 45,5 = 45

26. A = 29.05,5 = 60.95

www.ckl2.org 62

Chapter 10

TE Circles - TI

10.1 Chords and Circles

This activity is intended to supplement Geometry, Chapter 9, Lesson 3.

ID: 9773

Time required: 30 minutes

Topic: Circles

• Deduce from the Perpendicular Bisector Theorem the following corollaries:

a) The perpendicular from the center of a circle to a chord bisects the chord.

b) The line joining the center of a circle to the midpoint of a chord is perpendicular to the chord.

c) The center of a circle is at the intersection of the perpendicular bisector of two non-parallel chords.

Activity Overview

Students will begin this activity by exploring how the chord in a circle is related to its perpendicular bisector.
Investigation will include measuring lengths and distances from the center of the circle. These measurements
will then be transferred to a graph to see the locus of the intersection point of the measurements as the
endpoint of a chord is moved around the circle. In the extension, students will be asked to find an equation
for the ellipse that models the relationship.

Teacher Preparation

This activity is designed to be used in a high school geometry classroom.

• Students should already be familiar with circles, chords of circles, and perpendicular bisectors.
Classroom Management

• This activity is designed to be student- centered with the teacher acting as a facilitator while students
work cooperatively. Use the following pages as a framework as to how the activity will progress.

• Information for an optional extension is provided at the end of this activity.

63 www.ckl2.org

Software/Detail? id=258#.

Associated Materials

• Student Worksheet: Chords and Circles http://www.ckl2.org/flexr/chapter/9694

• Cabri Jr. Application

Introduction

When hikers and skiers go into terrain where there is a risk of avalanches, they take safety equipment
including avalanche rescue beacons. An avalanche rescue beacon sends and receives electromagnetic field
signals that can travel up to about 30 meters. The search pattern used to locate a beacon buried in the
snow is based on the properties of chords and diameters in circles. In this activity, you will use Cabri Jr.
to model an avalanche search pattern.

Problem 1 — Relationship between a chord an its perpendicular
bisector

Construct a circle using the Circle (F2 >Circle) too in Cabri Jr. to represent the beacon signal.

Use the Hide/Show to hide its center.

Construct a chord to represent the path of a rescuer as he walks a straight path until the signal disappears.
Use the Segment tool to draw the chord with endpoints on the circle and label it.

The rescuer walks back to the midpoint of this path. Find the midpoint of AB and label it M.

Construct a line perpendicular to AB through M, to represent the rescuer walking away from the path at
a 90° angle until the signal disappears.

Find one intersection point of the perpendicular line and the circle. Label it C.

The rescuer turns around and walks in the opposite direction until the signal disappears again. Find the
other intersection point of the perpendicular line and the circle. Label it D.

Hide the perpendicular line. Construct a segment connecting points C and D.

The rescuer walks back to the midpoint of this new path.

Find the midpoint of CD and label it X. This will be the center of the circle formed by the beacon signals.
Dig for the missing person!

Confirm that you have located the center of the circle. Measure the distances from X to A, B, C, and D.
www.ckl2.org 64

Problem 2 — Extension

Extension 1

Write a proof of the relationship used in the activity. Given a chord of a circle and its perpendicular
bisector, prove that the perpendicular bisector passes through the center of the circle.

Extension 2

Use a compass and straightedge to construct a circle and a chord. Construct the perpendicular bisector of
the chord and see that it passes through the center of the circle.

10.2 Inscribed Angles Theorem

This activity is intended to supplement Geometry, Chapter 9, Lesson 4-

ID: 12437

Time Required: 15 minutes

Activity Overview

Students will begin this activity by looking at inscribed angles and central angles and work towards discovering
a relationship among the two, the Inscribed Angle Theorem. Then, students will look at two corollaries to
the theorem.

Topic: Circles

• Construct central and inscribed angles

• Inscribed angles theorem

Teacher Preparation and Notes

• This activity was written to be explored with the Cabri Jr. application on the TI-84-

Software/Detail? id=258#.

US/Activities/Detail?id=12437 and select INSCRIB1-INSCRIB5.

Associated Materials

• Student Worksheet: Inscribed Angle Theorem http://www.ckl2.org/flexr/chapter/9694, scroll
down to the second activity.

• Cabri Jr. Application

. INSCRIB1.8xv, INSCRIB2.8xv, INSCRIB3.8xv, INSCRIB4.8xv, and INSCRIB5.8xv

Problem 1 — Similar Triangles

Students will begin this activity by looking at inscribed angles and central angles and work towards
discovering a relationship among the two.

65 www.ckl2.org

Students will be asked to collect data by moving points A and C. Students are asked questions about the
relationships in the circle and are asked to make a conjecture. In order to calculate the ratio of mlACB to

Q

-* M

jf\ *

.■fc£i

*J*

■ft

angled 73.30*

On page 1.6, students will look at two inscribed angles intercepted by the same arc and are asked to make

In an advanced setting a proof of the inscribed angle theorem and the two conjectures in problem one are
appropriate and can be proved using isosceles triangles.

Problem 2 — Extension of the Inscribed Angle Theorem

In Problem 2, students will look at two more angles created from the central angle and the intercepted
arc. Both sections of this problem are corollaries of the Inscribed Angle Theorem and both solutions are
congruent to the measure of the central angle intercepted by the arc or one-half the measure of the central
angle.

f Jr V* a lie

^ — """ FinGLEtefi.fiS*

priijLE['E,3. sir-

Solutions

Table 10.1:

Position

Measure of lACB

mlACB

1

2
3

4

36.65°
49.50°
49.50°
49.50°

73.30°
99.01°
99.01°
99.01°

0.5
0.5
0.5
0.5

2 i

www.ckl2.org

66

Table 10.2:

Position

Measure of LACB

Measure of lAEB

1
2
3
4

36.65°
48.30°
48.30°
48.30°

36.65°
48.30°
48.30°
48.30°

4. Sample answer: They are congruent.

5. diameter of the circle

6. 90°

Table 10.3:

Position

Measure of LACB

Measure of lAGE

1
2
3

4

36.65°
67.04°
67.04°
67.04°

73.30°
134.10°
134.10°
134.10°

36.65°
67.04°
67.04°
67.04°

Table 10.4:

Position

Measure of LACB

Measure of LABE

1

2
3

4

57.65°
49.20°
54.90°
63.55°

115.30°
98.40°
109.80°
127.10°

57.64°
49.19°
54.90°
63.54°

10.

10.3 Circle Product Theorems

This activity is intended to supplement Geometry, Chapter 9, Lesson 6.

ID: 12512

Time Required: 20 minutes

Activity Overview

Students will use dynamic models to find patterns. These patterns are the Chord- Chord, Secant-Secant,
and Secant- Tangent Theorems.

67

www.ckl2.org

Topic: Circles

• Chord- Chord, Secant- Secant, and the Secant- Tangent Product Theorems
Teacher Preparation and Notes

• This activity was written to be explored with the Cabri Jr. application on the TI-84-

Software/Detail? id=258#.

US/Activities/Detail?id=12512 and select PRODUC1-PRODUC5.

Associated Materials

• Student Worksheet: Circle Product Theorems http://www.ckl2.org/flexr/chapter/9694, scroll
down to the third activity.

• Cabri Jr. Application

. PR0DUC1.8xv, PRODUC2.8xv, and PRODUC3.8xv

Problem 1 — Chord-Chord Product Theorem

Students will begin this activity by investigating the intersection of two chords and the product of the
length of the segments of one chord and the product of the length of the segments of the other chord.

Students will be asked to collect data by moving point A. Students are asked to calculate the products
by hand on their accompanying worksheet. Students are asked several questions about the relationship
among the products.

As an extension, prove the chord-chord product theorem using similar triangles.

J^-

r

r

i

^E

*

f

---■^

DO

5

\

AK2.70

^-_ ^j

SKG.SS

CKi.Ei

DK i.7B

Problem 2 — Secant-Secant Product Theorem

Students will investigate the intersection of two secants and the product of the lengths of one secant
segment and its external segment and the product of the lengths of the other secant segment and its
external segment.

Students will be asked to collect data by moving point A. They are to calculate the products by hand
on their accompanying worksheet. Students are asked several questions about the relationship among the
products.

As an extension, prove the secant-secant product theorem using similar triangles.
www.ckl2.org 68

f r-

■-T--

fiK E.9H

C« 6.65
DK 3. 42

Problem 3 — Secant-Tangent Product Theorem

Students will investigate the intersection of the product of the lengths of one secant segment and its external
segment and the square of the tangent segment.

Students will be asked to collect data by moving point A. Students are asked to calculate the products
by hand on their accompanying worksheet. Students are asked several questions about the relationship
among the products.

As an extension, prove the secant-tangent product theorem using similar triangles.

(

CK fi.13
M 2.90

Problem 4 — Application of the Product Theorems

Students will be asked to apply what they learned in Problems 1-3 to solve a few problems.

Solutions

Table 10.5:

Position

AX

BX

cx

DX

AX-

BX

CXDX

1

2.70

0.99

1.51

1.78

2.67

2.68

2

2.51

1.05

1.38

1.90

2.63

2.62

3

2.93

0.91

1.80

1.49

2.67

2.68

4

1.80

1.08

2.51

0.77

1.94

1.93

2. They are equal.

3. equal

69 www.ckl2.org

Table 10.6:

Position AX

BX

CX

DX

AXBX

CXDX

1 5.94

3.82

6.65

3.42

22.69

22.74

2 4.85

3.16

5.85

2.62

15.33

15.33

3 4.03

2.75

5.32

2.09

11.08

11.12

4 7.47

4.96

7.92

4.68

37.05

37.07

5. They are equal.

6. equals

Table 10.7:

Position AX

CX

DX

AX 2

CXDX

1 4.22

6.13

2.90

17.81

17.77

2 3.42

5.40

2.16

11.70

11.66

3 2.45

4.56

1.32

6.00

6.02

4 9.46

11.21

7.98

89.49

89.46

8. They are equal.

9. equals

10. 6

11 2

±±. 4

12. 3Vl3or 10.817

www.ckl2.org TO

Chapter 11

TE Perimeter and Area - TI

11.1 Diameter and Circumference of a Circle

This activity is intended to supplement Geometry, Chapter 10, Lesson 4-

ID: 9844

Time required: 30 minutes

Topic: Circles

• Use technology to verify the circumference and area formulas for the circle.

Activity Overview

In this activity, students explore the relationship between a circle's circumference and its diameter. This
will lead students to their own discovery of a value for pi.

Teacher Preparation

This activity is designed to be used in a high school geometry clssroom.

• Students should already be familiar with circles, diameter, circumference, and pi.
Classroom Management

• This activity is designed to be student- centered with the teacher acting as a facilitator while students
work cooperatively. Use the following pages as a framework as to how the activity will progress.

Software/Detail?id=258#.

Associated Materials

• Student Worksheet: Diameter and Circumference of a Circle http : //www . ckl2 . org/f lexr/chapter/
9695

• Cabri Jr. Application

71 www.ckl2.org

In this activity we will

• Draw a circle

• Measure the diameter of the circle

• Measure the circumference of the circle

• Calculate the ratio of the circumference to the diameter.

Problem 1

Press APPS. Move down to the Cabri Jr. APP and press ENTER. Press ENTER, or any key, to begin
using the application.

Press Y = for the F\ menu and select New.

(If asked to Save changes? press <— ENTER to choose "No.")

Press WINDOW for the F2 menu, move down to Circle, and press ENTER. Press ENTER to mark
the center of the circle, then move the pencil to indicate the length of the radius, and press ENTER to
complete the circle.

Draw a line through the two points which determined the circle. To do this, press WINDOW for the F2
menu, move to Line, then press ENTER. Move the pencil until the point on the circle is flashing, and
press ENTER. Now move the pencil until the center of the circle is flashing, and press ENTER. Press
CLEAR to exit the line drawing tool.

\.

4

k /

Press WINDOW for F2 and move to Point. Move to the right and down to select Intersection. Press
ENTER. Move the pencil until both the line and the circle are flashing. Press ENTER to mark the
point which is the intersection of the circle and the line. Now we have two points on the circle which are
the endpoints of a diameter

To measure the circle's diameter, press GRAPH for F5 and move down and right to select Measure, D.
&: Length. Press ENTER.

Move the pencil until one endpoint of the diameter is flashing then press ENTER. Move to the other
endpoint of the diameter and when it is flashing, press ENTER. Press + to see the measurement rounded
to hundredths. The hand is active so you can move the measurement to a convenient location then press
ENTER.

The Measurement tool is still active so now you can find the circumference of the circle. Move the pencil
until the circle is flashing. Press ENTER then + to see the circumference rounded to hundredths. Move
the hand until the measurement is in a convenient location. Press ENTER. Press CLEAR to turn off
the measurement tool.

Press GRAPH for F5 and move down to Calculate. Press ENTER. Move the arrow until the circum-
ference measurement shows a flashing underline and press ENTER then -K Move the arrow until the
diameter measurement has a flashing underline and press ENTER again. The number displayed is the
ratio of the circle's circumference to its diameter.

www.ckl2.org 72

To explore this relationship with other circles, press CLEAR to turn off the Calculate tool. Move the
arrow until the point which defined the circle's radius or its center is flashing. Press ALPHA to activate
the hand. Grab the point and move it to change the size of the circle.

To confirm that the ratio is still 3.14, repeat the Calculate procedure. (It is actually being recalculated
each time the circle changes, but it is impossible to tell this since the number is unchanging.)

To exit the APP, press Y = for the Fl menu. Move to Quit, then press ENTER.

11.2 From the Center of the Polygon

This activity is intended to supplement Geometry, Chapter 10, Lesson 6.

ID: 11644

Time Required: 45 minutes

Activity Overview

Students will explore the area of a regular polygon in terms of the apothem and the perimeter. They will
derive the formula for a regular pentagon and regular hexagon. Then, students will see how the formula
relates to the formula for the area of triangles. Students will then be asked to apply what they have learned
about the area of a regular polygon.

• Regular Polygons

• Area of Regular Polygons

Teacher Preparation and Notes

• This activity was written to be explored with the Cabri Jr. app on the TI-84-

Software/Detail? id=258#.

US/Activities/Detail?id=11644 and select PENTAGON, HEXAGON, OCTAGON.

Associated Materials

• Student Worksheet: From the Center of a Polygon http://www.ckl2.org/flexr/chapter/9695,
scroll down to the second activity.

• Cabri Jr. Application

. PENTAG0N.8xv, HEXAG0N.8xv, 0CTAG0N.8xv

Problem 1 — Area of a Regular Pentagon

Students will begin this activity by looking at a regular pentagon. In file PENTAGON, students are given
regular pentagon ABCDE with center 7?. Students are given the length of CD (side of the polygon), RM
(apothem), and the area of the polygon. Students are to collect data in the table given on their student
worksheet and contains 4 columns: Apothem, Perimeter, a ■ p (apothem times perimeter), and
Area.

73 www.ckl2.org

Students will collect data by moving point D. They will do this for four different positions of the point.
Students are asked about how the area and the apothem times the perimeter are related.

F:Mi.

72

.j

K

CD ZSO
HF:Ert i0.r=:

t

~~1

^

E

*

\

/

\ /

cb— Am-iJ[i

Problem 2 — Area of a Regular Hexagon

Students will repeat the same process as Problem 1 for a regular hexagon. Students will begin to discover
the formula for a regular polygon is one-half the perimeter times the apothem.

Problem 3 — Area of a Regular Polygon

In this problem, students are to "prove" the formula for the area of a regular polygon by looking at an
octagon and the triangles created by the radii of the octagon. If students are confused by the term radius
of the polygon, explain that this is the radius of a circle circumscribed about a regular polygon.

Problem 4 — Area of Regular Polygons

In Problem 4, students are asked to apply what they have learned about the area of regular polygons The
students are given a question on the area of a regular polygon and a calculator on each page. The students
are to use the calculator to find the area.

Solutions

Table 11.1:

Position

Apothem (a)

Perimeter

(P)

a ■ p
times
ter)

(apothem
perime-

Area

1

1.72

12.5

21.5

10.75

2

1.93

14

27.02

13.49

3

2.13

15.5

33.015

16.53

4

2.55

18.5

47.175

23.55

2. area = ^ (apothem) (perimeter)

\ap

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74

Table 11.2:

Position

Apothem (a)

Perimeter

(p)

a • p
times
ter)

(apothem
perime-

Area

1

1.91

13.2

25.2

12.6

2

2.25

15.6

35.1

17.76

3

2.34

16.2

37.908

18.9

4

2.60

18

46.8

23.4

4. Area = I (apothem) (perimeter) = ^ap

5. 10

6. Yes

7. Yes

8. Area CDR = -as

9. Area = 2 as (&) = ^ as

10. Area = -^asn

11. 485.52 sq. in.

12. 25.2 sq. cm

13. 501.84 sq. ft

14. 1,039.2 sq. mm

T5 www.ckl2.org

Chapter 12

TE Surface Area and Volume
TI

12.1 Surface Area of a Cylinder

This activity is intended to supplement Geometry, Chapter 11, Lesson 2.

ID: 10075

Time required: 30 minutes

Activity Overview

Students will explore a net representation for a right cylinder. The surface area will be developed from the
parts of the net.

Topic: 3-Dimensional Geometry

• Construct 3- dimensional prisms and pyramids from nets.

• Calculate the surface area of a right prism or cylinder.

Teacher Preparation and Notes

• This activity is designed to be used in a high school or middle school geometry classroom.

• This activity is designed to be student- centered.

• The surface area of a right cylinder with base radius = R and height = H is SA = 2nR 2 + 2nRH. The
activity asks students to notice that the circumference of the circle is the length of the rectangle in
the net.

• The points R and H control the radius and the height of the cylinder. When R is dragged, the length
of the rectangle also changes (because the length = circumference of the circle). The height of the
rectangle does not change when R is dragged.

• Note: Measurements can display 0, 1, or 2 decimal digits. If digits are displayed, the value shown
will round from the actual value. To change the number of digits displayed:

1. Move the cursor over the value so it is highlighted.

2. Press + to display additional decimal digits or - to decrease digits.

www.ckl2.org T6

Software/Detail?id=258#.

US/Activities/Detail?id=10075 and select CYLINDER.

Associated Materials

• Student Worksheet: Surface Area of Cylinders http://www.ckl2.org/flexr/chapter/9696

• Cabri Jr. Application
. CYLINDER. 8xv

Problem 1 — Nets

A net is a pattern that can be cut out and folded into a 3- dimensional figure. Students should see a partial
net of a right cylinder. If the rectangle of the net were rolled up, the circle would be the top face of the
cylinder (like a lid of a jar).

?•%— °— f i.oo

Ah

i.PO

The dimensions of the net can be changed by dragging the points R and H. Students should drag these
points and notice what changes with the figure for each point. When R is moved, the radius of the circle
and the width of the rectangle are changed.

Students should use the D. & Length tool (F5 > Measure) to find the length of the rectangle.

Note: Display measurements with 2 decimal digits. To do this, hover the cursor over the measurement and
then press the plus key (+).

Next, they should use the Calculate tool to divide the length of the rectangle by the radius of the circle
to find that the width of the rectangle is the same as the circumference of the circle.

Problem 2 — Surface Area

Students should use the Area tool from the Measure menu (F5 > Measure) to find the areas of the
rectangle and the circle. Then, they should use the Alph-Num tool to place the number 2 on the screen.

Note: Press the ALPHA button to access numerical characters. The tool icon in the corner of the screen
will display A .

Press the ENTER button to start and end the text.

,..-—-■, :=.iH i

R^— °— \$ ij)0

S.2H

i H i

2. SO

i

iE.ro

II www.ckl2.org

Finally, students are to find the surface area of the cylinder. Remind students that this is only a partial
net, so one of the faces is missing. To find the surface area, students need to find the area of the circle and
the area of the rectangle. They should first use the Area tool from the Measurement menu to calculate
these areas. Then, they need to use the Calculate tool to find the sum of both circle bases by clicking on
the area of the circle, pressing I, and clicking on the 2. Next, they should click on the number they just
calculated, press +, and click on the area of the rectangle.

www.ckl2.org 78

Chapter 13

TE Transformations - TI

13.1 Transformations with Lists

This activity is intended to supplement Geometry, Chapter 12, Lessons 2 and 3.

ID: 10277

Time required: 40 minutes

Activity Overview

Students will graph a figure in the coordinate plane. They will use list operations to perform reflections,
rotations, translations and dilations on the figure, and graph the resulting image using a scatter plot.

Topic: Transformational Geometry

• Perform reflections, rotations, translations and dilations using lists and scatter plots to represent
figures on a coordinate plane.

Teacher Preparation and Notes

This activity is designed to be used in a high school geometry or algebra classroom.

If an original point on the coordinate plane is denoted by (x,y), then each of the following ordered

pairs denotes a transformation:

(x, —y) reflect over x — axis
(— x,y) reflect over y — axis
(y, x) reflect over y = x

(— y,x) rotate 90° around origin
{—x, —y) rotate 180° around origin
(y, -x) rotate 90° around origin

To perform a translation, add or subtract a constant from the list with the x— values or the y— values

of the figure.

To perform a dilation, multiply a constant scale factor by the list with the x- values or the y- values

of the figure.

This activity is designed to be student- centered with the teacher acting as a facilitator while students

work cooperatively. If desired, have students work in groups of 3. Each person in the group should

enter a different combination of lists for Problem 2 and the group should discuss the results.

79

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Associated Materials

Student Worksheet: Translations with Lists http://www.ckl2.org/flexr/chapter/9697

Problem 1 — Creating a Scatter Plot

Before beginning the activity, students need to clear all entries from the Y = screen and all lists.

First, students will enter the data on the worksheet into lists L\ and L2.

After setting up Plotl for a scatter plot of L\ vs L2 and changing the window settings, students view the
graph and sketch the figure on the worksheet. The shape should be an arrow, in the first quadrant, pointing
to the right.

Problem 2 — Reflections and Rotations

In the list editor students are to enter the formulas = —L\ and = — L-i for L3 and L4 respectively. This will
allow them to create several different reflections and rotations of the original figure.

To type Li, students need to press 2 [1].

To type L2, students need to press 2 nd [2].

For each combination of lists, students are to determine what type of reflection occurred.

A: x <— L3 and v <— L2

...<£

k>

(— x,y) over y-axis
B: x <— L\ and y <— L4

k?>

i\$

(x, —y) over x-axis
C: x <— L2 and y <— L\

www.ckl2.org

80

M>

(y, x) over the line y = x

Use Plot2 to create the following scatter plots. For each combination, determine what type of rotation
occurred.

D: x <— Li and y <— L\

tr : ^>

(— y, x) 90° around origin
E: x <— L2 and y <— L3

h>

: '"••■-■•■""

(-x, -y) 90° around origin
F: x <— L3 and y <— L4

fe>...

<£

(y, -x) 180° around origin

Problem 3 — Translations

In the list editor, students are to enter the formulas = L\ — 5 and = L2 + 3 for L3 and L4 respectively. This
will allow them to translate the original figure.

Students should see that the image shifted to the left 5 units and up 3 units. Remind students that the

81

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tick marks on the graph are every 2 units.

L

£>...

Now students are to translate the scatter plot into Quadrant 3 by editing the formula bars for L3 and L4.
Possible formulas are below.

L3 formula: = L\ — 15
The image shifted 15 units to the left and 10 units down.

L4 formula: = hi — 10

te>

&

Extension — Dilations

Remind students that they have seen dilations in Chapter 7, Lesson 6.

In the list editor, students are to enter the formulas = 0.5*Li and = 0.5*L2 for L3 and L4 respectively. This
will allow them to dilate the original figure.

Students should see that the image decreased in size. If they have trouble seeing the image, they can the
mark of the plot to the small dot.

Then students are to dilate the scatter plot into Quadrant 3 by editing the formula bars for L3 and L4.
Remind students that the scale factor needs to be the same for both lists. Possible formulas are below.

L3 formula: = -0.5*Li
L4 formula: = -0.5*L2

13.2 Reflections and Rotations

This activity is intended to supplement Geometry, Chapter 12, Lesson 5.

www.ckl2.org 82

ID: 8180

Time required: 45 minutes

Activity Overview

Students draw a puppet whose arms and legs are constructed by using a translation vector and reflection
line. They will manipulate the vector and observe the changes in the puppet. This activity will help students
visualize the effect of a transformation in their construction.

Topic: Transformational Geometry

• Given a translation vector and a geometric figure, translate the figure to discover that lengths, angles,
areas, shapes and parallel line segments are preserved under translations.

• Given a reflection line and a geometric figure, reflect the figure to discover that lengths, angles, areas
and shapes are preserved under reflections and orientations are reversed.

Teacher Preparation and Notes

• Although this is a geometry activity, it can also be used in a Pre-Algebra or Algebra classroom.

• Before beginning this activity, students should be familiar with reflections and translations outside
the coordinate plane. They should already know the terms associated with transformations, such as
line of reflection and translation vector.

• This activity is designed to be explored individually or in pairs, with the teacher acting as a facilitator.

Software/Detail? id=258#.

Associated Materials

• Student Worksheet: Reflections <fe Rotations http://www.ckl2.org/flexr/chapter/9697, scroll
down to the second activity.

• Cabri Jr. Application

Problem 1 — Constructing the puppet

They will use the Circle, Segment, and Point On tools to create the head, body, and shoulder, respec-
tively.

Students must draw the segment for the translation vector in the order described on the worksheet (bottom
right point, then top left point).

83 www.ckl2.org

They are directed to use the Translation tool to create the left hand and then use the Reflection tool
to create the right hand. Each point will need to be joined with a segment to the shoulder point.

Another segment to be used as the translation vector for the legs will be drawn with the first one. Again,
students must draw the segment in the order described on the worksheet (top right point, then bottom
left point).

Problem 2— Moving the puppet

Students should drag the point joining the two translation vectors and observe the changes of the arms
and legs.

J>

Ni

To investigate the puppet further, students should use the Slope tool to find the slopes of the arms. Then
they can drag the point on the left side of the screen and observe the changes.

Students can also find the coordinates of the left and right hands.

^

y\,

13.3 Perspective Drawing

This activity can be used as an end of the year activity or to supplement the "Know What?"
from Chapter 7, Lesson 6.

ID: 10033

Time required: 60 minutes

Activity Overview

In this activity, students draw figures in one- and two-point perspective and compare and contrast the two
types of drawings. They then create an isometric drawing and compare it to their drawings in perspective.

Topic: 3— Dimensional Geometry

• Construct 3- dimensional prisms and pyramids.

• Record the number of faces, edges, and vertices of prisms.

Teacher Preparation and Notes

www.ckl2.org 84

• Perspective drawings can be taught at any time in a geometry curriculum, but are most appropriate
after lessons on parallel and perpendicular lines, three-dimensional figures, and symmetry.

• Throughout the activity, students use many drawing and construction tools, such as the Segment,
Parallel, and Perpendicular tools. In this document, the first use of a tool is by name and accom-
panied by its location within the menu structure. For subsequent uses, the tool may be mentioned by
name or its function, and the menu location is omitted.

• This activity is designed to have students explore individually or in pairs.

Software/Detail? id=258#.

US/Activities/Detail?id=10033 and select RECPRISM, TRIPRISM, TWOPERSP, ISO-
DRAW.

Associated Materials

• Student Worksheet: Perspective Drawing http : //www . ckl2 . org/f lexr/chapter/9697, scroll down
to the third activity.

• Cabri Jr. Application

. RECPRISM.8xv, TRIPRISM.8xv, TWOPERSP.8xv, and ISODRAW.8xv

Problem 1 — One-point perspective

Students will open the Cabri Jr. file RECPRISM and find a vaishing point A along the horizon and a
rectangle BCDE.

They are to use the Segment tool from the F2 menu to draw AB and AC. These two segments (and any
other segment that joins a vertex of the prism to the vanishing point) are hereafter referred to as vanishing
segments.

Next, they will create FG such that F is between A and B, G is between A and C, and FG x BC (Refer to
the diagram at the right).

To do this, they will need to do the following:

. Place a point F on AB. (F2 > Point > Point On)

Construct a line through F parallel to BC.
(F3 > Parallel)

• Plot point G at the intersection of the parallel line and AC. (F2 > Point > Intersection)

• Hide the line. (F5 > Hide/Show > Object)

• Draw FG.

Students can now hide the vanishing segments, AB and AC, and draw BF and CG. The front and top faces
of the rectangular prism have been constructed.

Ask students if they are satisfied that this drawing completely represents a rectangular prism. Have them
drag point A (press APLHA to grab the point) to either the far left or far right and ask students what is
wrong with the drawing. (The rectangular prism does not have a face on either side.)

To complete the prism, students next need to construct AD and AE. Then, using the Perpendicular tool
(F3 > Perp.), construct two lines perpendicular to FG — one through F and the other through G. The

85 www.ckl2.org

vertical "back" edges GH and FJ may then be created by plotting the intersection points, hiding the lines,
and constructing the segments as needed.

To show all six edges, students should draw a segment connecting H and /. Once all of the vanishing
segments are hidden, students will find they need to also construct segments DH and EJ.

H

-a-

& 1

Depending on the topics you have recently covered in class, you can now have students identify parallel,
intersecting, and skew lines; count faces, vertices, and edges; or discuss the similarity of rectangles BCDE
and FGHJ.

When students have finished exploring their prisms, they may wish to hide or dash the edges that would
not be visible if the figure were not transparent. The appearance of the edges may be altered using the
Display tool (F5 > Display).

Students are to open TRIPRISM and create a triangular prism in one-point perspective on their own.
Be sure to circulate around the room and assist students as needed.

Allow students a few minutes to explore the figure by dragging point A along the horizon or dragging an
edge. Vertices B, C, and D can also be dragged on this figure.

Problem 2 — Two-point perspective

The Cabri Jr. file TWOPERSP contains two vanishing points, A and B, signifying this rectangular prism
will be drawn in two-point perspective. The file also contains a vertical line segment, CD, which will be
the front edge of the prism.

CD is constructed so that dragging point C changes the segment's length and dragging point D translates
the segment.

Next, students will then use the Segment tool to draw the vanishing segments AC, AD, BC, and BD.
Then they should use the Point On, Parallel, Intersection, Hide/Show, and Segment tools (as they
did in Problem 1) to construct the vertical edges, EF and GH, of the left and right faces.

The top and bottom edges of those faces, CE, CG, DF, and DH may be drawn as well.

Students should now hide the four vanishing segments, and then draw four more: from E and F to B and
from G and H to A.

The intersection of the upper two vanishing segments and that of the lower two vanishing segments should
be plotted and a segment drawn between them to form the last vertical edge, JK.

www.ckl2.org 86

The vanishing segments may then be hidden so that students can draw the remaining edges of the prism.

As before, allow students some time to drag edges, vertices, and vanishing points, observing how the prism
changes as they do so. (Dragging CD is very interesting!) Again, students can either hide or dash any
"hidden edges."

Problem 3 — An isometric drawing

Using ISODRAW, students will construct an isometric drawing of a rectangular prism. The three seg-
ments shown represent the three "front" edges of the prism, and each of the angles the segments form
measures 120°. Students should first use the Parallel, Intersection, Hide/Show, and Segment tools
to construct AE, CF , DE, and DF, as shown in the diagram to the right.

They may then use the same sequence of tools once more to construct the remaining edges: AG, CG, EH,
FH, and GH.

I

"7i

After completing the isometric drawing, students should compare it to the perspective drawings from
Problems 1 and 2. As a similarity, students may identify that right angles "in real-life" do not actually
measure 90° in the drawings. As a difference, all parallel lines "in real-life" are parallel in an isometric
drawing, whereas in perspective drawings, some parallel lines actually "meet" at the vanishing point.

87

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```