11.4 Arcs and Chords - Murrieta Valley Unified School District

Page 1 of 5

11.4 Arcs and Chords

Goal

Use properties of chords of circles.

Key Words

? congruent arcs p. 602 ? perpendicular bisector

p. 274

By finding the perpendicular bisectors of two chords, an archaeologist can recreate a whole plate from just one piece.

This approach relies on Theorem 11.5, and is shown in Example 2.

THEOREM 11.4

Words If a diameter of a circle is perpendicular

to a chord, then the diameter bisects the chord and its arc.

Symbols If B&G* F&D*, then D&E* c E&F* and DsG c GsF .

B

F E

G D

EXAMPLE 1 Find the Length of a Chord

In C the diameter A&F* is perpendicular to B&D*.

D

Use the diagram to find the length of B&D*. Solution

5

F

C

E

A

Because A&F* is a diameter that is perpendicular to B&D*,

you can use Theorem 11.4 to conclude that A&F* bisects

B

B&D*. So, BE ED 5.

BD BE ED 55 10

Segment Addition Postulate Substitute 5 for BE and ED. Simplify.

ANSWER The length of B&D* is 10.

Find the Length of a Segment

1. Find the length of J&M*. H

12

K

CM

G J

608 Chapter 11 Circles

2. Find the length of S&R*.

P

C

S

N

15 P

R

Careers

Page 2 of 5

THEOREM 11.5

Words If one chord is a perpendicular bisector

of another chord, then the first chord is

a diameter.

J

Symbols If J&K* M&L* and M&P** c P&L*, then J&K* is

a diameter.

M PK L

All diameters of a circle include the center of the circle. Therefore, the point where two diameters intersect is the center of the circle.

EXAMPLE 2 Find the Center of a Circle

Suppose an archaeologist finds part of a circular plate. Show how to reconstruct the original shape of the plate.

Solution

1 Draw any two chords that are not

parallel to each other.

ARCHAEOLOGISTS study and reconstruct artifacts which provide information about past cultures.

Career Links



2 Draw the perpendicular bisector of

each chord. These lines contain diameters.

3 The diameters intersect at the circle's

center. Use a compass to draw the rest of the plate.

Student Help

STUDY TIP If two central angles are congruent then their corresponding arcs are congruent.

THEOREM 11.6

Words In the same circle, or in congruent circles:

? If two chords are congruent, then their

corresponding minor arcs are congruent. A

B

? If two minor arcs are congruent, then their

corresponding chords are congruent.

D

C

Symbols If A&B* c D&C*, then AsB c DsC .

If AsB c DsC , then A&B* c D&C*.

11.4 Arcs and Chords 609

Page 3 of 5

EXAMPLE 3 Find Measures of Angles and Chords

Find the value of x.

a.

S

P

R

60

x C

P

b. A

70 B

x

C 3E

Solution

D 70

a. Because Q&P* c R&S*, it follows that QsP c RrS. So, mQsP mRrS 60, and x 60.

b. Because ArB c DsE , it follows that A&B* c D&E*. So, x DE 3.

Find Measures of Angles and Chords

Find the value of x.

3.

4.

A

E

C

4

x

B

D

H x 2

J

G

5

F

5.

U

V

40

W

Y

(x 10)

Z

11.4 Exercises

Guided Practice

Vocabulary Check

1. Identify a diameter.

C B

A

F

GD E

2. Identify a pair of congruent arcs. P P C

S

R

Skill Check

Find the value of x.

3.

R

S6 TC

x 1

P

P

610 Chapter 11 Circles

4.

M

(x 5)

H

N

50 JK

5. A

D 8

B

Cx

E

Page 4 of 5

Practice and Applications

Extra Practice

See p. 695.

Identifying Diameters Determine whether A&B* is a diameter of the circle. Explain your reasoning.

6.

C

A

E B 85

7.

A

CE 5

D 6

8.

C

4

E

A

B

4

D

B

D

Finding Chords and Central Angles Find the value of x.

9. A D

x

7

C

B

E

10.

A

11.

B C 35

D E (x 6)

A x7

C

B

D 3x 1

Logical Reasoning Name any congruent arcs, chords, or angles. State a postulate or theorem that justifies your answer.

12.

B

13.

A

90

14. A

C

P

A

C

B

P

C

90

D

P

B

D

Finding Measures Find the measure of the red segment or arc.

15.

B

DA

10

C

E

16. B

40

A

D

17.

B

E

110 AC

F C

E 60 D

Homework Help

Example 1: Exs. 6?8, 15?20

Example 2: Exs. 21?22 Example 3: Exs. 9?20

Using Algebra Find the value of x.

18. B (2x 10) A

C

D E (x 30)

19. D

A

x 6

C

E

3x 8

B

20.

A

(x 45)

C

B

4x

D

E

21. Visualize It! Draw a large circle and cut it out. Tear part of it off

and ask another student to recreate your circle.

11.4 Arcs and Chords 611

Page 5 of 5

Careers

22. Avalanche Rescue Beacon An avalanche rescue beacon is a device used by backcountry skiers. It gives off a signal that is detectable within a circle of a certain radius. In a practice drill, a ski patrol uses the following steps to locate a beacon buried in the snow. Explain how it works.

1 Walk in a straight line until the signal disappears. Turn around and walk back until the signal disappears again.

2 Walk back to the halfway point, and walk away from the line at a 90 angle until the signal disappears.

EMTS Some Emergency Medical Technicians (EMTs) train specifically for wilderness emergencies.

Career Links



starting point

3 Turn around and walk in a straight line until the signal disappears again.

4 Walk back to the halfway point. You will be near the center of the circle. The beacon is under you.

hidden beacon

Standardized Test Practice

23. Multi-Step Problem Use the diagram below.

a. Explain why AsD c BsE .

A 40 E

b. Find the value of x. c. Find mAsD and mBsE . d. Find mBsD.

(15x 40) 5 C

D

5 (10x 10) B

Mixed Review Algebra Skills

Measuring Arcs In the diagram below, A&D* and B&E* are diameters

of F. Find the measure. (Lesson 11.3)

24. mDsE

25. mBsC

26. mAsE

27. mBt CD

E F A 40 65 D

28. mAs BC

29. mAs DE

B C

Comparing Numbers Compare the two numbers. Write the answer using , or . (Skills Review, p. 662)

30. 26 and 29

31. 1250 and 34

32. 0.2 and 15

612 Chapter 11 Circles

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download