Using Chords 10.3

10.3 Using Chords

EEssentialssential QQuestionuestion What are two ways to determine when a chord is a diameter of a circle?

Drawing Diameters Work with a partner. Use dynamic geometry software to construct a circle of radius 5 with center at the origin. Draw a diameter that has the given point as an LOOKING FOR endpoint. Explain how you know that the chord you drew is a diameter. STRUCTURE a. (4, 3) b. (0, 5) c. (−3, 4) d. (−5, 0) To be profi cient in math, you need to look closely to discern a pattern Writing a Conjecture about Chords or structure. Work with a partner. Use dynamic geometry— software to construct a C chord BC of a circle A. Construct a chord— on the perpendicular bisector of BC . What do you notice? Change the original chord and the circle several times. Are your results A always the same? Use your results to write a conjecture. B

A Chord Perpendicular to a Diameter — Work with a partner. Use dynamic geometry software to construct a diameter BC A DE— BC— F of a circle . Then construct a chord perpendicular to at point . Find— the lengths DF and EF. What do you notice? Change the chord perpendicular to BC and the circle several times. Do you always get the same results? Write a conjecture about a chord that is perpendicular to a diameter of a circle.

F A E C

CCommunicateommunicate YourYour AnswerAnswer 4. What are two ways to determine when a chord is a diameter of a circle?

Section 10.3 Using Chords 545

hhs_geo_pe_1003.indds_geo_pe_1003.indd 545545 11/19/15/19/15 2:332:33 PMPM 10.3 Lesson WWhathat YYouou WWillill LLearnearn Use chords of circles to fi nd lengths and arc measures.

Core VocabularyVocabulary Using Chords of Circles Previous Recall that a chord is a segment with endpoints semicircle major arc chord on a circle. Because its endpoints lie on the arc circle, any chord divides the circle into two arcs. diameter A diameter divides a circle into two semicircles. diameter chord Any other chord divides a circle into a minor arc and a major arc. semicircle minor arc READING TTheoremsheorems GD២ ≅ GF២ If , then the point Theorem 10.6 Congruent Corresponding Chords Theorem G, and any line, segment, or ray that contains G, In the same circle, or in congruent circles, two B C bisects FD២ . minor arcs are congruent if and only if their corresponding chords are congruent. F AD E ២ ២ G Proof Ex. 19, p. 550 AB ≅ CD if and only if AB ≅ CD.

D Theorem 10.7 Perpendicular Chord Bisector Theorem EG FD២ bisects . If a diameter of a circle is perpendicular F to a chord, then the diameter bisects the chord and its arc. E H G

D EG EG ⊥ DF If is a diameter and២ ២ , Proof Ex. 22, p. 550 then HD ≅ HF and GD ≅ GF .

Theorem 10.8 Perpendicular Chord Bisector Converse T If one chord of a circle is a perpendicular bisector of another chord, then the fi rst chord S is a diameter. P Q

R If QS is a perpendicular bisector of TR, Proof Ex. 23, p. 550 then QS is a diameter of the circle.

Using Congruent Chords to Find an Arc Measure — — In the diagram, ⊙P ≅ ⊙Q, FG ≅ JK , J mJK២ = ° mFG២ and 80 . Find . ° P 80 Q G F K SOLUTION FG— JK— Because ២ and ២are congruent chords in congruent circles, the corresponding minor arcs FG and JK are congruent by the Congruent Corresponding Chords Theorem. ២ ២ So, mFG = mJK = 80°.

546 Chapter 10 Circles

hhs_geo_pe_1003.indds_geo_pe_1003.indd 546546 11/19/15/19/15 2:332:33 PMPM Using a Diameter J (70 + x)° ២ K a. Find HK. b. Find mHK . 11x° N 7 SOLUTION H M — — JL HK L a. Diameter is perpendicular— to . So,— by the Perpendicular Chord Bisector Theorem, JL bisects HK , and HN = NK. So, HK = 2(NK) = 2(7) = 14. JL— HK— b. Diameter — is perpendicular២ to ២ . So, ២by the Perpendicular Chord Bisector Theorem, JL bisects HK , and mHJ = mJK . ២ ២ m HJ = mJK Perpendicular Chord Bisector Theorem 11x° = (70 + x)° Substitute. 10x = 70 Subtract x from each side. x = 7 Divide each side by 10. mHJ២ = mJK២ = + x ° = + ° = ° So,២ ២ (70 ) (70 7) 77 , and mHK = 2(mHJ ) = 2(77°) = 154°.

Using Perpendicular Bisectors

Three bushes are arranged in a garden, as shown. Where should you place a sprinkler so that it is the same distance from each bush?

SOLUTION Step 1 Step 2 Step 3

BC BC BC

A A A sprinkler

Label the bushes A, B, Draw the perpendicular— — Find the point where the and C, as shown.— Draw— bisectors of AB and BC . perpendicular bisectors segments AB and BC . By the Perpendicular Chord intersect. This is the center Bisector Converse, these lie of the circle, which is on diameters of the circle equidistant from points A, containing A, B, and C. B, and C.

MMonitoringonitoring ProgressProgress Help in English and Spanish at BigIdeasMath.com ⊙D In Exercises 1 and 2, use the diagram of . B ២ ២ A 9 1. If mAB = 110°, fi nd mBC . ២ ២ D 2. If mAC = 150°, fi nd mAB . 9 C In Exercises 3 and 4, fi nd the C indicated length or arc measure. x° B 5 9 3. CE A F D ២ 4. mCE E (80 − x)°

Section 10.3 Using Chords 547

hhs_geo_pe_1003.indds_geo_pe_1003.indd 547547 11/19/15/19/15 2:332:33 PMPM TTheoremheorem Theorem 10.9 Equidistant Chords Theorem In the same circle, or in congruent circles, two chords C G are congruent if and only if they are equidistant from A the center. E D F B Proof Ex. 25, p. 550 AB ≅ CD if and only if EF = EG.

Using Congruent Chords to Find a Circle’s Radius

W R In the diagram, QR = ST = 16, CU = 2x, and CV = 5x − 9. Find the radius of ⊙C. U SOLUTION Q 2x CQ— C Because is a segment whose— endpoints— are the center and a point on the circle, 5x − 9 it is a radius of ⊙C. Because CU ⊥ QR , △QUC is a right triangle. Apply properties Y of chords to fi nd the lengths of the legs of △QUC. S V T X W R U

Q 2x C radius 5x − 9 Y S V T X

Step 1 Find CU. — — — — Because QR and ST are congruent chords, QR and ST are equidistant from C by the Equidistant Chords Theorem. So, CU = CV. CU = CV Equidistant Chords Theorem 2x = 5x − 9 Substitute. x = 3 Solve for x. So, CU = 2x = 2(3) = 6. Step 2 Find QU. — — — — Because diameter WX ⊥ QR , WX bisects QR by the Perpendicular Chord Bisector Theorem. 1 QU = — = So, 2 (16) 8.

Step 3 Find CQ. Because the lengths of the legs are CU = 6 and QU = 8, △QUC is a right triangle with the Pythagorean triple 6, 8, 10. So, CQ = 10. J R P So, the radius of ⊙C is 10 units.

3x K N Help in English and Spanish at BigIdeasMath.com 7x − 12 MMonitoringonitoring ProgressProgress T JK = LM = NP = x NQ = x − L Q M 5. In the diagram, 24, 3 , and 7 12. Find the ⊙N S radius of .

548 Chapter 10 Circles

hhs_geo_pe_1003.indds_geo_pe_1003.indd 548548 11/19/15/19/15 2:332:33 PMPM 10.3 Exercises Dynamic Solutions available at BigIdeasMath.com

VVocabularyocabulary andand CoreCore ConceptConcept CheckCheck 1. WRITING Describe what it means to bisect a chord.

2. WRITING Two chords of a circle are perpendicular and congruent. Does one of them have to be a diameter? Explain your reasoning. MMonitoringonitoring ProgressProgress andand ModelingModeling withwith MathematicsMathematics

In Exercises 3–6, fi nd the measure of the red arc or 12. PROBLEM SOLVING In the cross section of the chord in ⊙C. (See Example 1.) submarine shown, the control panels are parallel and the same length. Describe a method you can use 3. A E 4. T to fi nd the center of the cross section. Justify your 34° method. (See Example 3.) 75° 5 U C C 34° B D V

5. W 6. L 120° 11 ° 110 X M C 7 P Q N Z — 120° C In Exercises 13 and 14, determine whether AB is a ° 60 Y diameter of the circle. Explain your reasoning. 7 R A S 13. C 14. B 5 3 In Exercises 7–10, fi nd the value of x. (See Example 2.) C D D 3 E 7. J 8. U A F 8 B C C x E H In Exercises 15 and 16, fi nd the radius of ⊙Q. G R T (See Example 4.) x S 40° A 15. E 16 16. x + B G F 4 4 9. M 5x − 6 10. A D F 4x + 3 L Q Q 5 Q 5 H 7x − 6 D N B P H G 6x − 6 2x + 9 C (5x + 2)° E (7x − 12)° 16

17. PROBLEM SOLVING An archaeologist fi nds part of a 11. ERROR ANALYSIS Describe and correct the error circular plate. What was the diameter of the plate to in reasoning. the nearest tenth of an inch? Justify your answer. ✗ A 7 in. B Because AC— — 7 in. E bisects DB , 6 in. BC២ ≅ CD២ . 6 in. C D Section 10.3 Using Chords 549

hhs_geo_pe_1003.indds_geo_pe_1003.indd 549549 11/19/15/19/15 2:332:33 PMPM 18. HOW DO YOU SEE IT? What can you conclude 22. PROVING A THEOREM Use congruent triangles to from each diagram? Name a theorem that justifi es prove the Perpendicular Chord Bisector Theorem your answer. (Theorem 10.7). — D E a. D b. A Given EG is a diameter of ⊙L. L A 90° — — EG ⊥ DF C G E B Q — — ២ ២ Prove DC ≅ FC , DG ≅ FG F C C B D 23. PROVING A THEOREM Write a proof of the 90° Perpendicular Chord Bisector Converse c. J d. P (Theorem 10.8). T S — Given QS is a perpendicular P F N RT— S Q Q Q bisector of . — L L Prove QS is a diameter of R H R L G M the circle . (Hint: Plot the center L and draw △LPT and △LPR.) 19. PROVING A THEOREM Use the diagram to prove each part of the biconditional in the Congruent 24. THOUGHT PROVOKING A C Corresponding Chords Theorem (Theorem 10.6). Consider two chords that intersect at point P. Do you AP CP P A — = — think that BP DP? Justify D P B your answer. D B C 25. PROVING A THEOREM Use the diagram with the AB— CD— a. Given ២ and២ are congruent chords. Equidistant Chords Theorem (Theorem 10.9) on Prove AB ≅ CD page 548 to prove both parts of the biconditional of this theorem. AB២ ≅ CD២ b. Given — — Prove AB ≅ CD 26. MAKING AN ARGUMENT A car is designed so that the rear wheel is only partially visible below the body 20. MATHEMATICAL CONNECTIONS A of the car. The bottom edge of the panel is parallel to In ⊙P, all the arcs shown have the ground. Your friend claims that the point where x AB២ integer measures. Show that x° C the tire touches the ground bisects . Is your friend must be even. P correct? Explain your reasoning.

21. REASONING In ⊙P, the lengths of the parallel chords២ are 20, 16, and 12. Find mAB . Explain P your reasoning. A B A B

MMaintainingaintaining MathematicalMathematical ProficiencyProficiency Reviewing what you learned in previous grades and lessons Find the missing interior angle measure. (Section 7.1) 27. Quadrilateral JKLM has angle measures m∠J = 32°, m∠K = 25°, and m∠L = 44°. Find m∠M. 28. Pentagon PQRST has angle measures m∠P = 85°, m∠Q = 134°, m∠R = 97°, and m∠S = 102°. Find m∠T.

550 Chapter 10 Circles

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