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11.4 Arcs and Chords

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

This approach relies on Theorem 11.5, and is Key Words shown in Example 2. • congruent arcs p. 602 • perpendicular bisector p. 274 THEOREM 11.4 B Words If a diameter of a circle is perpendicular to a chord, then the diameter bisects the F chord and its arc. E G Symbols If BG&* ∏ FD&* , then DE&* c EF&* and DGs c GFs. D

EXAMPLE 1 Find the Length of a Chord In ᭪C the diameter AF&* is perpendicular to BD&* . D Use the diagram to find the length of BD&* . 5 F C Solution E A Because AF&* is a diameter that is perpendicular to BD&* , you can use Theorem 11.4 to conclude that AF&* bisects B BD&* . So, BE ϭ ED ϭ 5.

BD ϭ BE ϩ ED Segment Addition Postulate ϭ 5 ϩ 5 Substitute 5 for BE and ED . ϭ 10 Simplify.

ANSWER ᮣ The length of BD&* is 10.

Find the Length of a Segment

1. Find the length of JM&* . 2. Find the length of SR&* . H P 12 K C S C M N G 15 P J R

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THEOREM 11.5

Words If one chord is a perpendicular bisector M of another chord, then the first chord is a diameter. J P K Symbols If JK&* ∏ ML&* and MP&** c PL&* , then JK&* is a diameter. 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. Careers 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.

study ARCHAEOLOGISTS ●2 Draw the perpendicular bisector of and reconstruct artifacts each chord. These lines contain which provide information about past cultures. diameters. Career Links CLASSZONE.COM

●3 The diameters intersect at the circle’s center. Use a compass to draw the rest of the plate.

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 Student Help B • If two minor arcs are congruent, then their STUDY TIP D corresponding chords are congruent. If two central angles C are congruent then Symbols If AB&* c DC&* , then ABs c DCs. their corresponding arcs are congruent. If ABs c DCs, then AB&* c DC&* .

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EXAMPLE 3 Find Measures of Angles and Chords Find the value of x. a. S b. 70 ؇ A B P R x ؇ 60 x؇ C C P 3 E

D 70 ؇ Solution a. Because QP&* c RS&* , it follows that QPs c RSr. So, mQPs ϭ mRSr ϭ 60 Њ, and x ϭ 60. b. Because ABr c DEs, it follows that AB&* c DE&* . So, x ϭ DE ϭ 3.

Find Measures of Angles and Chords

Find the value of x.

3. 4.H 5. U V x ؉ 2 40 ؇ A E C W J G x 4 5 Y B D Z F (x ؉ 10) ؇

11.411.4 Exercises Guided Practice

Vocabulary Check 1. Identify a diameter. 2. Identify a pair of congruent arcs. C P B P A F C G D E S R

Skill Check Find the value of x. 3.R 4.M 5. A x ؉ ؇ ( 5) D S 6 H N 8 T C B x C x ؊ 1 ؇ P P 50 E J K

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Practice and Applications

Extra Practice Identifying Diameters Determine whether AB&* is a diameter of the See p. 695. circle. Explain your reasoning. 6. 7.A 8. C C A C E D 4 E 5 6 E A B ؇ 85 B 4

D B D

Finding Chords and Central Angles Find the value of x. 9.A 10.A 11. A D x ؉ 7 B x C 35 ؇ 7 C C B D (x ؉ 6) ؇ B E E 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 14. A ؇ C 90 B P P P C B D A C 90 ؇ D

Finding Measures Find the measure of the red segment or arc. 15.B 16. 17. B B ؇ ؇ A E 110 D A 40 A C D 10 F E C E C ؇ 60 D

Using Algebra Find the value of x. 18. 19. 20. B (2 x ؊ 10) ؇ A A A (x ؉ 45) ؇ x ؉ 6 C C C D E B D Homework Help 3x ؊ 8 4x؇ E (x ؉ 30) ؇ Example 1: Exs. 6–8, B D E 15–20 Example 2: Exs. 21–22 21. Example 3: Exs. 9–20 Visualize It! Draw a large circle and cut it out. Tear part of it off and ask another student to recreate your circle.

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22. Avalanche Rescue Beacon An avalanche rescue beacon is a device used by backcountry skiers. It gives off a signal that is Careers 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 2 Walk back to the halfway point, disappears. Turn around and walk and walk away from the line at a 90 Њ back until the signal disappears again. angle until the signal disappears.

starting point

EMTS Some Emergency Medical Technicians (EMTs) 3 Turn around and walk in a straight line 4 Walk back to the halfway point. train specifically for wilderness until the signal disappears again. You will be near the center of the emergencies. circle. The beacon is under you. Career Links CLASSZONE.COM

hidden beacon

Standardized Test 23. Multi-Step Problem Use the diagram below. ؇ Practice a. Explain why ADs c BEs. A 40 E b. x (15 x ؊ 40) ؇ (10 x ؉ 10) ؇ Find the value of . 5 5 c. Find mADs and mBEs. C D B d. Find mBDs.

Mixed Review Measuring Arcs In the diagram below, AD&* and BE&* are diameters of ᭪F. Find the measure. (Lesson 11.3) E 24. mDEs 25. mBCs F A D mAEs 27. mBCDt 40 ؇ 65 ؇ .26 B 28. mABCs 29. mADEs C

Algebra Skills Comparing Numbers Compare the two numbers. Write the answer (Skills Review, p. 662) .؍ using <, >, or 15 3 1 30. Ϫ26 and Ϫ29 31. ᎏᎏ and Ϫᎏᎏ 32. 0.2 and ᎏᎏ 20 4 5

612 Chapter 11 Circles