Essential Understanding Theorem 12-4 and Its Conversewhslmeyer.weebly.com/uploads/5/8/1/8/58183903/12-2.pdfLesson 12-2 Chords and Arcs 775 Using Diameters and Chords Archaeology An

Lesson 12-2 Chords and Arcs 771

12-2

Objectives To use congruent chords, arcs, and central angles To use perpendicular bisectors to chords

Chords and Arcs

Lesson Vocabulary

•chord

LessonVocabulary

Theorem 12-4 and Its Converse

Theorem Within a circle or in congruent circles, congruent central angles have congruent arcs.

ConverseWithin a circle or in congruent circles, congruent arcs have congruent central angles.

If ∠AOB ≅ ∠COD, then AB¬ ≅ CD¬.

If AB¬ ≅ CD¬, then ∠AOB ≅ ∠COD.

You will prove Theorem 12-4 and its converse in Exercises 19 and 35.hsm11gmse_1202_t08234

B

AC

D

O

How can you use congruent triangles to help with this one?

}A @ }D, and jA @ jD. If BC = 15, what is the length of EF? How do you know? FD

E

C

B

A

In the Solve It, you found the leng th of a chord, which is a segment whose endpoints

are on a circle. The diagram shows the chord PQ and its related arc, PQ¬.

Essential Understanding You can use information about congruent parts of a circle (or congruent circles) to find information about other parts of the circle (or circles).

The following theorems and their converses confirm that if you know that chords, arcs, or central angles in a circle are congruent, then you know the other two parts are congruent.

hsm11gmse_1202_t06825.ai

Q

O

PMATHEMATICAL PRACTICES

G-C.A.2 Identify and describe relationships among inscribed angles, radii, and chords.

MP 1, MP 3

Common Core State Standards

Problem 1

772 Chapter 12 Circles

Using Congruent Chords

In the diagram, }O @ }P. Given that BC @ DF, what can you conclude?

∠O ≅ ∠P because, within congruent circles, congruent chords have

congruent central angles (conv. of Thm. 12-5). BC¬ ≅ DF¬ because, within

congruent circles, congruent chords have congruent arcs (Thm. 12-6).

1. Reasoning Use the diagram in Problem 1. Suppose you are given }O ≅ }P and ∠OBC ≅ ∠PDF. How can you show ∠O ≅ ∠P?

From this, what else can you conclude?


Theorem Within a circle or in congruent circles, congruent central angles have congruent chords.

ConverseWithin a circle or in congruent circles, congruent chords have congruent central angles.

If ∠AOB ≅ ∠COD, then AB ≅ CD.

If AB ≅ CD, then ∠AOB ≅ ∠COD.

You will prove Theorem 12-5 and its converse in Exercises 20 and 36.


Theorem Within a circle or in congruent circles, congruent chords have congruent arcs.

ConverseWithin a circle or in congruent circles, congruent arcs have congruent chords. If AB ≅ CD, then AB¬ ≅ CD¬.

If AB¬ ≅ CD¬, then AB ≅ CD.

You will prove Theorem 12-6 and its converse in Exercises 21 and 37.

hsm11gmse_1202_t08235

B

A

C

DO

hsm11gmse_1202_t08236

B

A

C

DO


CO

B D

PF

Got It?


Theorem Within a circle or in congruent circles, chords equidistant from the center or centers are congruent.

ConverseWithin a circle or in congruent circles, congruent chords are equidistant from the center (or centers).

If OE = OF, then AB ≅ CD.

If AB ≅ CD, then OE = OF.

You will prove the converse of Theorem 12-7 in Exercise 38.hsm11gmse_1202_t08242.ai

B

A

D

C

FE

O

Why is it important that the circles are congruent?Two circles may have central angles with congruent chords, but the central angles will not be congruent unless the circles are congruent.

Problem 2


Proof of Theorem 12-7

Given: }O, OE ≅ OF, OE # AB, OF # CD

Prove: AB ≅ CD

Statements Reason

1) OA ≅ OB ≅ OC ≅ OD 1) Radii of a circle are congruent.

2) OE ≅ OF, OE # AB, OF # CD 2) Given

3) ∠AEO and ∠CFO are right angles. 3) Def. of perpendicular segments

4) △AEO ≅ △CFO 4) HL Theorem

5) ∠A ≅ ∠C 5) Corres. parts of ≅ △s are ≅.

6) ∠B ≅ ∠A, ∠C ≅ ∠D 6) Isosceles Triangle Theorem

7) ∠B ≅ ∠D 7) Transitive Property of Congruence

8) ∠AOB ≅ ∠COD 8) If two ⦞ of a △ are ≅ to two ⦞ of another △, then the third ⦞ are ≅.

9) AB ≅ CD 9) ≅ central angles have ≅ chords.

Finding the Length of a Chord

What is the length of RS in }O?

PQ = QR = 12.5 Given in the diagram

PQ + QR = PR Segment Addition Postulate

12.5 + 12.5 = PR Substitute.

25 = PR Add.

RS = PR Chords equidistant from the center of a circle are congruent.

RS = 25 Substitute.

2. What is the value of x? Justify your answer.

Proof

hsm11gmse_1202_t06827

A

O

BE

FC

D

hsm11gmse_1202_t06828

PQ

O

S

R9

9

12.5

PR ≅ RS, since they are the same distance from the center of the circle. So finding PR gives the length of RS.

The length of chord RS

The diagram indicates that PQ = QR = 12.5 and PR and RS are both 9 units from the center.


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9876543210

9876543210

9876543210

9876543210

2 5

9876543210

−

hsm11gmse_1202_t06829

x 36

1816

18Got It?


The Converse of the Perpendicular Bisector Theorem from Lesson 5-2 has special applications to a circle and its diameters, chords, and arcs.

Proof of Theorem 12-9

Given: }O with diameter AB bisecting CD at E

Prove: AB # CD

Proof: OC = OD because the radii of a circle are congruent. CE = ED by the definition of bisect. Thus, O and E are both equidistant from C and D. By the Converse of the Perpendicular Bisector Theorem, both O and E are on the perpendicular bisector of CD. Two points determine one line or

segment, so OE is the perpendicular bisector of CD. Since OE is part of AB, AB # CD.

Theorem 12-8

TheoremIn a circle, if a diameter is perpendicular to a chord, then it bisects the chord and its arc.

If . . .AB is a diameter and AB # CD

Then . . .

CE ≅ ED and CA¬ ≅ AD¬

You will prove Theorem 12-8 in Exercise 22.

Theorem 12-9

TheoremIn a circle, if a diameter bisects a chord (that is not a diameter), then it is perpendicular to the chord.

If . . .AB is a diameter and CE ≅ ED

Then . . .AB # CD

Theorem 12-10

TheoremIn a circle, the perpendicular bisector of a chord contains the center of the circle.

If . . .AB is the perpendicular

bisector of chord CD

Then . . .AB contains the center of }O

You will prove Theorem 12-10 in Exercise 33.


C

EA

D

BO


C

EA

D

BO


C

A

D

BEO


C

A

D

BEO


C

A

D

B


C

A

D

BO

Proof

hsm11gmse_1202_t06830

AO

E B

D

C

Problem 3

Problem 4


Using Diameters and Chords

Archaeology An archaeologist found pieces of a jar. She wants to find the radius of the rim of the jar to help guide her as she reassembles the pieces. What is the radius of the rim?

Step 1 Trace a piece of the rim. Draw two chords and construct perpendicular bisectors.

Step 2 The center is the intersection of the perpendicular bisectors. Use the center to find the radius.

hsm11gmse_1202_t08246 hsm11gmse_1202_t082475

43

21

0inch

6

The radius is 4 in.

3. Trace a coin. What is its radius?

Finding Measures in a Circle

Algebra What is the value of each variable to the nearest tenth?

A LN = 12(14) = 7 A diameter # to a chord bisects the chord.

r 2 = 32 + 72 Use the Pythagorean Theorem.

r ≈ 7.6 Find the positive square root of each side.

B BC # AF

A diameter that bisects a chord that is not a diameter is # to the chord.

BA = BE = 15 Draw an auxiliary BA. The auxiliary BA ≅ BE

because they are radii of the same circle.

y2 + 112 = 152 Use the Pythagorean Theorem.

y2 = 104 Solve for y 2.

y ≈ 10.2 Find the positive square root of each side.

4. Reasoning In part (b), how does the auxiliary BA make the problem simpler to solve?

Got It?

hsm11gmse_1202_t06831

L N M

K

14 cm

r3 cm

hsm11gmse_1202_t06832

AE

F

BC y11

15

11

Got It?

How does the construction help find the center?The perpendicular bisectors contain diameters of the circle. Two diameters intersect at the circle’s center.

Find two sides of a right triangle. The third side is either the answer or leads to an answer.


Practice and Problem-Solving Exercises

In Exercises 6 and 7, the circles are congruent. What can you conclude?

6. 7.

Find the value of x.

8. 9. 10.

11. In the diagram at the right, GH and KM are perpendicular bisectors of the chords they intersect. What can you conclude about the center of the circle? Justify your answer.

12. In }O, AB is a diameter of the circle and AB # CD. What conclusions can you make?

See Problem 1.PracticeA

hsm11gmse_1202_t06834

A

X

YZ

B

C


T

H

N

FG

EJ

M

L

K

See Problem 2.


Ox5

57


x

8

15


x

5

3.5

hsm11gmse_1202_t08251

H M

K G See Problems 3 and 4.


C

BEA

D

O

Lesson CheckDo you know HOW?

In }O, mCD¬ = 50 and CA @ BD.

1. What is mAB¬? How do you know?

2. What is true of CA¬ and BD¬? Why?

3. Since CA = BD, what do you know

about the distance of CA and BD from the center of }O?

Do you UNDERSTAND? 4. Vocabulary Is a radius a chord? Is a diameter a

chord? Explain your answers.

5. Error Analysis What is the error in the diagram?

hsm11gmse_1202_t08249

AB

D

O

C

hsm11gmse_1202_t08250

Q

PS

R

4

43

3

MATHEMATICAL PRACTICES

MATHEMATICAL PRACTICES


Algebra Find the value of x to the nearest tenth.

13. 14. 15.

16. Geometry in 3 Dimensions In the figure at the right, sphere O with radius 13 cm is intersected by a plane 5 cm from center O. Find the radius of the cross section }A.

17. Geometry in 3 Dimensions A plane intersects a sphere that has radius 10 in., forming the cross section }B with radius 8 in. How far is the plane from the center of the sphere?

18. Think About a Plan Two concentric circles have radii of 4 cm and 8 cm. A segment tangent to the smaller circle is a chord of the larger circle. What is the length of the segment to the nearest tenth?

• How will you start the diagram? • Where is the best place to position the radius of each circle?

19. Prove Theorem 12-4.

Given: }O with ∠AOB ≅ ∠COD

Prove: AB¬ ≅ CD¬


Given: }O with AB ≅ CD

Prove: AB¬ ≅ CD¬

} A and } B are congruent. CD is a chord of both circles.

23. If AB = 8 in. and CD = 6 in., how long is a radius?

24. If AB = 24 cm and a radius = 13 cm, how long is CD?

25. If a radius = 13 ft and CD = 24 ft, how long is AB?

26. Construction Use Theorem 12-5 to construct a regular octagon.


O

x20

16


Ox

3.6

8


x

12

6


5 cm 13 cmO

A

ApplyB

Proof

hsm11gmse_1202_t08252

AO

BC

D

Proof


CA

B

D

O


C

BA

D


Given: }O with ∠AOB ≅ ∠COD

Prove: AB ≅ CD

Proof


CA

B D

O


Given: }O with diameter ED # AB at C

Prove: AC ≅ BC, AD¬ ≅ BD¬

Proof


C

A

B

E DO


27. In the diagram at the right, the endpoints of the chord are the points where the line x = 2 intersects the circle x2 + y2 = 25. What is the length of the chord? Round your answer to the nearest tenth.

28. Construction Use a circular object such as a can or a saucer to draw a circle. Construct the center of the circle.

29. Writing Theorems 12-4 and 12-5 both begin with the phrase, “within a circle or in congruent circles.” Explain why the word congruent is essential for both theorems.

Find mAB¬. (Hint: You will need to use trigonometry in Exercise 32.)

30. 31. 32.


Given: / is the # bisector of WY.

Prove: / contains the center of }X.

Prove each of the following.

35. Converse of Theorem 12-4: Within a circle or in congruent circles, congruent arcs have congruent central angles.

36. Converse of Theorem 12-5: Within a circle or in congruent circles, congruent chords have congruent central angles.

37. Converse of Theorem 12-6: Within a circle or in congruent circles, congruent arcs have congruent chords.

38. Converse of Theorem 12-7: Within a circle or congruent circles, congruent chords are equidistant from the center (or centers).

39. If two circles are concentric and a chord of the larger circle is tangent to the smaller circle, prove that the point of tangency is the midpoint of the chord.


y

xO�3

x � 2

�3

3

3


C

A

O

BD

16

16108�


A B

O


BO

A

3017

Proof


W Y

X

Z

�

ChallengeCProof

Proof

34. Given: }A with CE # BD

Prove: BC¬ ≅ DC¬

Proof


D

A

BB

E F

C


Mixed Review

Assume that the lines that appear to be tangent are tangent. O is the center of each circle. Find the value of x to the nearest tenth.

44. 45.

46. The legs of a right triangle are 10 in. and 24 in. long. The bisector of the right angle cuts the hypotenuse into two segments. What is the length of each segment, rounded to the nearest tenth?

Get Ready! To prepare for Lesson 12-3, do Exercises 47–52.

Identify the following in }P at the right.

47. a semicircle 48. a minor arc 49. a major arc

Find the measure of each arc in }P.

50. ST¬ 51. STQ¬ 52. RT¬

See Lesson 12-1.


O 140� x�


O

3.5

2

6.8x

See Lesson 7-5.

See Lesson 10-6.


SR

QP

T

86�145�

Standardized Test Prep

40. The diameter of a circle is 25 cm and a chord of the same circle is 16 cm. To the nearest tenth, what is the distance of the chord from the center of the circle?

9.0 cm 18.0 cm

9.6 cm 19.2 cm

41. The Smart Ball Company makes plastic balls for small children. The diameter of a ball is 8 cm. The cost for creating a ball is 2 cents per square centimeter. Which value is the most reasonable estimate for the cost of making 1000 balls?

$2010 $16,080

$4021 $42,900

42. From the top of a building you look down at an object on the ground. Your eyes are 50 ft above the ground and the angle of depression is 50°. Which distance is the best estimate of how far the object is from the base of the building?

42 ft 65 ft

60 ft 78 ft

43. A bicycle tire has a diameter of 17 in. How many revolutions of the tire are necessary to travel 800 ft? Show your work.

SAT/ACT

ShortResponse

Documents

Essential Understanding Theorem 12-4 and Its Conversewhslmeyer.weebly.com/uploads/5/8/1/8/58183903/12-2.pdfLesson 12-2 Chords and Arcs 775 Using Diameters and Chords Archaeology An