dg4te 883 06.qxd 10/24/06 6:51 pm page 317 chord …math.kendallhunt.com/.../dg4te_lesson6.2.pdf ·...

LESSON 6.2 Chord Properties 317

L E S S O N

6.2You will do foolish things, but

do them with enthusiasm.

SIDONIE GABRIELLA COLETTE

L E S S O N

6.2

Investigation 1Defining Angles in a CircleWrite a good definition of each boldfaced term. Discuss your definitions with

others in your group. Agree on a common set of definitions as a class and add

them to your definition list. In your notebook, draw and label a figure to illustrate

each term.

Step 1 Central Angle.

�AOB, �DOA, and �DOB �PQR, �PQS, �RST, �QST, and

are central angles of circle O. �QSR are not central angles of circle P

R

Q

P

S

T

D

A

B

O

Step 2 Inscribed Angle.

�ABC, �BCD, and �CDE are �PQR, �STU, and �VWX are

inscribed angles. not inscribed angles.

Q

PV

R

T

S UX

W

A

B

E

D

C

Chord PropertiesIn the last lesson you discovered some properties of a tangent, a line that intersects

the circle only once. In this lesson you will investigate properties of a chord, a line

segment whose endpoints lie on the circle.

In a person with correct vision, light rays from

distant objects are focused to a point on the

retina. If the eye represents a circle, then the path

of the light from the lens to the retina represents

a chord. The angle formed by two of these chords

to the same point on the retina represents an

inscribed angle.

First you will define two types of angles in a circle.

LESSON OBJECTIVES

� Discover properties of chords to a circle� Practice construction skills

NCTM STANDARDS

CONTENT PROCESS

Number � Problem Solving

Algebra � Reasoning

� Geometry � Communication

� Measurement � Connections

Data/Probability Representation

PLANNING

LESSON OUTLINE

One day:25 min Investigation

10 min Sharing

5 min Closing

5 min Exercises

MATERIALS

� construction tools� protractors� Gear Fragment (W) for One Step� Sketchpad activity Chord Properties,

optional� Sketchpad demonstration Intersecting

Tangents Conjecture, optional

TEACHING

In this lesson students discover

some properties relating central

angles, chords, and arcs of

circles. Begin with the one-step

investigation, or ask groups to

work through Investigations

1 through 4.

You can replace or extend

Investigations 1 and 2 with the

Dynamic Geometry Exploration

at www.keymath.com/DG.

You might have students use

patty paper for Investigations 2

through 4. You can have students

construct circles on patty paper

by tracing circular objects.

Guiding Investigation 1

One StepHand out the Gear Fragment

worksheet and pose this pro-

blem: “In repairing a machine,

you find a fragment of a circular

gear. To replace the gear, you

need to know its diameter.

How can you find the gear’s

Step 1 A central angle has its vertex at the center of the circle.

Step 2 An inscribed angle has its vertex on the circle and its sides are chords.

DG4TE_883_06.qxd 10/24/06 6:51 PM Page 317

Step 3 How can you fold your circle construction to check

the conjecture?

Step 4 Recall that the measure of an arc is defined as the

measure of its central angle. If two central angles are

congruent, their intercepted arcs must be congruent.

Combine this fact with the Chord Central Angles

Conjecture to complete the next conjecture.

“Pull the cord?! Don’t I needto construct it first?”

Chord Arcs Conjecture

If two chords in a circle are congruent, then their �? are congruent.

C-56

Chord Central Angles Conjecture

If two chords in a circle are congruent, then they determine two central

angles that are �?

C-55

? ?

Next you will discover some properties of chords and

central angles. You will also see a relationship between

chords and arcs.

Step 1 Construct a large circle. Label the center O. Construct

two congruent chords in your circle. Label the chords

AB� and CD�, then construct radii OA�, OB�, OC��, and OD��.

Step 2 With your protractor, measure �BOA and �COD. How do they compare? Share

your results with others in your group. Then copy and complete the conjecture.

AC

D

O

B

Investigation 2Chords and Their Central Angles

You will need

● a compass

● a straightedge

● a protractor

● patty paper (optional)

diameter?” You may need to

review the term diameter right

away. As you circulate, wonder

aloud as needed whether the

center is somehow related to the

chords of the circle, pointing out

what you mean by chord.

Step 2 If groups are having diffi-

culty defining inscribed angle,

[Ask] “Where have you heard

the word inscribed before?”

[A triangle inscribed in a circle

(whose center is the circum-

center of the triangle) has all

three vertices on the circle, and

its sides are chords of the circle.

An inscribed triangle has three

inscribed angles.]


[Alert] If you find that all or

most students in a group are

using a central angle that

measures 60° (because they

didn’t change the compass

setting after drawing the circle),

point out that they don’t have

enough information for good

inductive reasoning; then have

them all change their settings

from the circle’s radius and

begin again.

[Alert] Students may have trouble

constructing congruent chords.

Challenge them to find the

endpoints without using a ruler.

They might mark two points on

the circle and then use a

constant compass opening to

locate two other points on the

circle the same distance apart.

Step 3 If their paper is too thick

to see through when folded,

students may want to copy their

figure to patty paper.

Students may find this investiga-

tion and the next two investiga-

tions fairly straightforward

and may begin writing conjec-

318 CHAPTER 6 Discovering and Proving Circle Properties

intercepted arcs

Step 3 Fold it so that OD�� coincides with OB� and OC�� coincides with OA�.

congruent

tures without going through the steps or comparing

results with others in their group. Remind them

that the process is more important than the results.

If students finish early, you might challenge them

with the one-step problem.

DG4TE_883_06.qxd 10/24/06 6:51 PM Page 318

Investigation 3Chords and the Centerof the CircleYou will need

● a compass

● a straightedge


In this investigation you will discover relationships about a

chord and the center of its circle.

Step 1 Construct a large circle and mark the center. Construct

two nonparallel congruent chords. Then construct the

perpendiculars from the center to each chord.

Step 2 How does the perpendicular from the center of a circle

to a chord divide the chord? Copy and complete the conjecture.

Let’s continue this investigation to discover a relationship between the length of

congruent chords and their distances from the center of the circle.

Step 3 Compare the distances (measured along the perpendicular) from the center to

the chords. Are the results the same if you change the size of the circle and the

length of the chords? State your observations as your next conjecture.

Perpendicular to a Chord Conjecture

The perpendicular from the center of a circle to a chord is the �? of

the chord.

C-57

Chord Distance to Center Conjecture

Two congruent chords in a circle are �? from the center of the circle.

C-58

Next, you will discover a property of perpendicular

bisectors of chords.

Step 1 Construct a large circle and mark the center. Construct

two nonparallel chords that are not diameters. Then

construct the perpendicular bisector of each chord and

extend the bisectors until they intersect.

Investigation 4Perpendicular Bisectorof a ChordYou will need

● a compass

● a straightedge




Step 1 Students can use the same

two congruent chords and the

same drawing for Investigations

2 and 3. [Alert] Students may

need some review in how to

construct perpendiculars. Using

compass constructions to

complete all four investigations

may be too time-consuming, but

this construction and the next

can be completed quickly using

patty-paper constructions. The

perpendicular through the center

of the circle to the chord can be

folded.


The perpendicular bisectors of

the chords can be constructed by

simply folding the chord in half.

The Perpendicular Bisector of a

Chord Conjecture is the converse

of the Perpendicular to a Chord

Conjecture.

SHARING IDEAS

As usual, for presentations select

groups that have a variety of

statements of the conjectures. As

students present, encourage them

to put the ideas in their own

words, not just those of the

conjectures as presented in the

student book. Help the class

reach consensus on the wording

to record in their notebooks.

[Ask] “How would you define

congruent arcs?” Although the

measure of an arc was first

defined in Chapter 1, some

students may still be wondering

why arcs are measured in degrees

based on their central angle.

Drawing a central angle of 90°

may help some students relate

the central angle to a quarter of

a circle. But one source of diffi-

culty may be a natural tendency

to think that measure means

“size” in some direct way, and the

measure of an arc doesn’t give its

length. In fact, in different

circles, arcs with the same

Sharing Ideas (continued)measure can have very different lengths. Ask

whether having the same measure is enough to

make two arcs congruent, as is the case for line

segments and angles. [Ask] “What’s needed to

ensure that two arcs have the same size and shape?”

[Congruent arcs must be on the same or congruent

circles.] (Some students may think that an arc of a

larger circle will have the same size and shape as an

arc of a smaller circle. To the extent possible, let

other students convince them that the curvature

will be different.)

[Ask] “Congruent chords are equidistant from the

center. Can we say anything about distance from

the center if one chord is longer than the other?”

[In the same circle, shorter chords are farther from

the center.]

bisector

equidistant

DG4TE_883_06.qxd 10/24/06 6:51 PM 19

EXERCISESSolve Exercises 1–10. State which conjectures or definitions you used.

1. x � �? 2. z � �? 3. w � �?

4. AB � CD 5. AB�� is a diameter. Find 6. GIAN is a kite.

PO � 8 cm mAC� and m�B. Find w, x, and y.

OQ � �?

7. AB � 6 cm OP � 4 cm 8. mAC� � 130° 9. x � �?

CD � 8 cm OQ � 3 cm Find w, x, y, and z. y � �?

BD � 6 cm z � �?

What is the perimeter of

OPBDQ?

y

O x

AE

z

72�F Tw

y Ox

A z

D

B C

110�

48�

A

P

B

D

C

QO

Gy

x

w

I

A N

O

115�

C

68�

A

B

OAP

B

D

C

QO

70� w128�

20�

z

165�

x

�

You will need

Construction toolsfor Exercises 16–19

With the perpendicular bisector of a chord, you can find the center of any circle,

and therefore the vertex of the central angle to any arc. All you have to do is

construct the perpendicular bisectors of nonparallel chords.

Perpendicular Bisector of a Chord Conjecture

The perpendicular bisector of a chord �? .

C-59

Step 2 What do you notice about the point of intersection? Compare your results with

the results of others near you. Copy and complete the conjecture.

[� For interactive versions of these investigations, see the Dynamic Geometry Exploration Chord Propertiesat . �]www.keymath.com/DG

keymath.com/DG

|

� Helping with the Exercises

5. mAC� � 68°; m�B � 34° (Because �OBC is

isosceles, m�B � m�C, m�B � m�C � 68°, and

therefore m�B � 34°.)

Exercise 7 [Alert] Some students may neglect to add

on the length OQ.

Exercise 9 As needed, [Ask] “What do the measures

of all the arcs add up to?” [360°]


Assessing ProgressYou can assess students’ under-

standing of (and use of the vocab-

ulary for) radius, chord, central

angle, and inscribed angle and

their skill at constructing a circle,

measuring an angle with a

protractor, and comparing

segments with a compass. You

might also see how well they

understand the difference between

drawing and constructing.

Closing the Lesson

Summarize that the major conjec-

tures of this lesson are about

congruent chords of a circle: They

determine congruent central

angles, they intercept congruent

arcs, and they are equidistant

from the center. Another pair of

conjectures about chords forms a

biconditional: A line through the

center of a circle is perpendicular

to a chord if and only if it bisects

the chord. If students are still

shaky about these ideas, you

might want to use one of

Exercises 1–6 as an example.

BUILDINGUNDERSTANDING

The exercises focus on applying

the conjectures about chords.

Encourage students to sketch

pictures on their own papers.

They can then mark and label all

information accordingly.

ASSIGNING HOMEWORK

Essential 1–14, 17, 18

Performanceassessment 17, 18

Portfolio 20

Journal 13, 14, 16

Group 15, 21

Review 22–28

Algebra review 15, 21, 26

MATERIALS

� circular objects and patty paper(Exercise 17)

� Exercises 18 and 19 (T), optional

passes through the center of the circle

definition of measure of an arc Chord Arcs Conj. Chord Central Angles Conj.

Chord Distance to Center Conj. Chord Arcs Conjecture

w � 115°x � 115°y � 65°

Perpendicular to a Chord Conj. definition of arc measure

9. x � 96°, Chord Arcs Conjecture; y � 96°,

Chord Central Angles Conjecture; z � 42°,

Isosceles Triangle Conjecture and Triangle Sum

Conjecture.

165° 84°

8 cm

20 cm

70°

w � 110°x � 48°y � 82°z � 120°

DG4TE_883_06.qxd 10/24/06 6:51 PM Page 320

10. AB�� CO��, mCI� � 66° 11. Developing Proof What’s wrong 12. Developing Proof What’s wrong

Find x, y, and z. with this picture? with this picture?

13. Draw a circle and two chords of unequal length. Which is closer to the center of the

circle, the longer chord or the shorter chord? Explain.

14. Draw two circles with different radii. In each circle, draw a chord so that the chords

have the same length. Draw the central angle determined by each chord. Which

central angle is larger? Explain.

15. Polygon MNOP is a rectangle inscribed in a circle centered at the

origin. Find the coordinates of points M, N, and O.

16. Construction Construct a triangle. Using the sides of the triangle as

chords, construct a circle passing through all three vertices. Explain.

Why does this seem familiar?

17. Construction Trace a circle onto a blank sheet of paper

without using your compass. Locate the center of the circle

using a compass and straightedge. Trace another circle onto

patty paper and find the center by folding.

18. Construction Adventurer Dakota Davis digs up a piece of a

circular ceramic plate. Suppose he believes that some

ancient plates with this particular design have a diameter of

15 cm. He wants to calculate the diameter of the original

plate to see if the piece he found is part of such a plate.

He has only this piece of the circular plate, shown at right,

to make his calculations. Trace the outer edge of the plate

onto a sheet of paper. Help him find the diameter.

19. Construction The satellite photo at right shows only a portion

of a lunar crater. How can cartographers use the photo to

find its center? Trace the crater and locate its center. Using

the scale shown, find its radius. To learn more about satellite

photos, go to .www.keymath.com/DG

O37 cm

18 cm

A

C

I

O

Bx

y

z

P (4, 3)M (?, 3)

N (?, ?) O (?, ?)

0 1 2 3 4 5 km

Exercise 10 [Alert] Students may

miss the fact that the two radii in

�ABO are congruent, making it

an isosceles triangle.

10. x � 66°, y � 48°, z � 66°;

Corresponding Angles

Conjecture, Isosceles Triangle

Conjecture, Linear Pair

Conjecture

Exercise 12 If students are having

difficulty, [Ask] “What does the

perpendicular bisector have

to go through?” [the center of

the circle]

13. The longer chord is closer

to the center; the longest chord,

which is the diameter, passes

through the center.

14. The central angle of the

smaller circle is larger, because

the chord is closer to the center.

Exercise 16 As needed, remind

students of the meaning of a

triangle inscribed in a circle (or

a circle circumscribed around a

triangle).

16. The center of the circle is

the circumcenter of the triangle.

Possible construction:

5 cm

5 cm

Exercise 17 Have available round objects larger than

coins for tracing.

17. possible construction:

O

Exercises 17–19 If students are having difficulty,

wonder aloud whether there’s a conjecture that ends

with something about the center of a circle.

[Perpendicular Bisector of a Chord]

18. � 13.8 cm


The length ofthe chord isgreater than thelength of thediameter.

The perpendicularbisector of the segment does notpass through the center of the circle.

M(�4, 3), N(�4, �3), O(4, �3)

They can draw twochords and locate the intersection of their perpendicularbisectors. The radius is just over 5 km.

DG4TE_883_06.qxd 10/24/06 6:51 PM Page 321

21. Circle O has center (0, 0) and passes through points A(3, 4) and

B(4, �3). Find an equation to show that the perpendicular bisector

of AB� passes through the center of the circle. Explain your

reasoning.

Review

22. Developing Proof Identify each of these statements as true or false. If the statement is

true, explain why. If it is false, give a counterexample.

a. If the diagonals of a quadrilateral are congruent, but only one is the

perpendicular bisector of the other, then the quadrilateral is a kite.

b. If the quadrilateral has exactly one line of reflectional symmetry, then the

quadrilateral is a kite.

c. If the diagonals of a quadrilateral are congruent and bisect each other, then

it is a square.

23. Mini-Investigation Use what you learned in the last lesson about the angle formed by a

tangent and a radius to find the missing arc measure or angle measure in each

diagram. Examine these cases to find a relationship between the measure of the

angle formed by two tangents to a circle, �P, and the measure of the intercepted arc,

AB. Then copy and complete the conjecture below.

20. Developing Proof Complete the flowchart proof shown, which proves that if two

chords of a circle are congruent, then they determine two congruent central

angles.

Given: Circle O with chords AB� � CD�

Show: �AOB � �COD

Flowchart Proof

1 AB � CD

�?

2 AO � CO

BO � DO

4 � �? � � �

?

All radii of a circleare congruent

5

3

�?

�?

�AOB � � �?

�?

�

y

x

A (3, 4)

O

B (4, –3)

A

P Q y

B

20� P Q

A

x

B

40� 120� Q zP

A

B

O

A

B

C

D

Conjecture: The measure of the angle formed by two intersecting tangents

to a circle is �? (Intersecting Tangents Conjecture).

Exercise 23 You might use the

Sketchpad demonstration Inter-

secting Tangents Conjecture to

preview or replace this exercise.


Chapter 5

Given

CPCTCSSS CongruenceConjecture

All radii of a circle are congruent

�COD�AOB � �COD

y � �17

� x; (0, 0) is a point on this line.

false, isosceles trapezoid

true

false, rectangle

140°

160°

60°

180° minus the measure of the intercepted arc

22a.

Possible explanation: If AB� is the perpendicular

bisector of CD��, then every point on AB� is

equidistant from endpoints C and D. Therefore

C

X

D

BA

AC�� AD�� and BC�� BD�. Because CD�� is not the

perpendicular bisector of AB�, C is not equidistant

from A and B. Likewise, D is not equidistant from

A and B. So, AC� and BC� are not congruent, and

AD�� and BD� are not congruent. Thus ACBD

has exactly two pairs of consecutive congruent

sides, so it is a kite.

DG4TE_883_06.qxd 10/24/06 6:51 PM Page 322

dg4te 883 06.qxd 10/24/06 6:51 pm page 317 chord …math.kendallhunt.com/.../dg4te_lesson6.2.pdf ·...

Documents