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And Are Congruent Chords, So the Corresponding Arcs RS and ST Are Congruent

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And Are Congruent Chords, So the Corresponding Arcs RS and ST Are Congruent 9-3 Arcs and Chords ALGEBRA Find the value of x. 3. SOLUTION: 1. In the same circle or in congruent circles, two minor SOLUTION: arcs are congruent if and only if their corresponding Arc ST is a minor arc, so m(arc ST) is equal to the chords are congruent. Since m(arc AB) = m(arc CD) measure of its related central angle or 93. = 127, arc AB arc CD and . and are congruent chords, so the corresponding arcs RS and ST are congruent. m(arc RS) = m(arc ST) and by substitution, x = 93. ANSWER: 93 ANSWER: 3 In , JK = 10 and . Find each measure. Round to the nearest hundredth. 2. SOLUTION: Since HG = 4 and FG = 4, and are 4. congruent chords and the corresponding arcs HG and FG are congruent. SOLUTION: m(arc HG) = m(arc FG) = x Radius is perpendicular to chord . So, by Arc HG, arc GF, and arc FH are adjacent arcs that Theorem 10.3, bisects arc JKL. Therefore, m(arc form the circle, so the sum of their measures is 360. JL) = m(arc LK). By substitution, m(arc JL) = or 67. ANSWER: 67 ANSWER: 70 eSolutions Manual - Powered by Cognero Page 1 9-3 Arcs and Chords 5. PQ ALGEBRA Find the value of x. SOLUTION: Draw radius and create right triangle PJQ. PM = 6 and since all radii of a circle are congruent, PJ = 6. Since the radius is perpendicular to , bisects by Theorem 10.3. So, JQ = (10) or 5. 7. Use the Pythagorean Theorem to find PQ. SOLUTION: In the same circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. Therefore, 5x = 105 So, PQ is about 3.32 units long. x = 21 ANSWER: ANSWER: 3.32 21 6. In , GH = 9, KL = 4x + 1. Find x. 8. SOLUTION: SOLUTION: In the same circle or in congruent circles, two minor In the same circle or in congruent circles, two chords arcs are congruent if and only if their corresponding are congruent if and only if they are equidistant from the center. Since JS = JR, KL = GH. chords are congruent. Therefore, ANSWER: 70 ANSWER: 2 eSolutions Manual - Powered by Cognero Page 2 9-3 Arcs and Chords 11. 9. SOLUTION: SOLUTION: In the same circle or in congruent circles, two minor In the same circle or in congruent circles, two minor arcs are congruent if and only if their corresponding arcs are congruent if and only if their corresponding chords are congruent. Here, chords are congruent. So, Therefore, The sum of the measures of the central angles of a 3x + 5 = 26 circle with no interior points in common is 360. So, x = 7 ANSWER: 2x = 254 7 x = 127 ANSWER: 127 12. SOLUTION: In the same circle or in congruent circles, two minor 10. arcs are congruent if and only if their corresponding SOLUTION: chords are congruent. Here, In the same circle or in congruent circles, two minor Therefore, arcs are congruent if and only if their corresponding 5x – 1 = 4x + 3 chords are congruent. Therefore, x = 4 2x – 1 = 143 ANSWER: x = 72 4 ANSWER: 72 eSolutions Manual - Powered by Cognero Page 3 9-3 Arcs and Chords 13. 15. MODELING Angie is in a jewelry making class at her local arts center. She wants to make a pair of triangular earrings from a metal circle. She knows that is . If she wants to cut two equal parts off so that , what is x? SOLUTION: In the same circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. Here, Therefore, 5x = 3x + 54 x = 4 SOLUTION: ANSWER: In the same circle or in congruent circles, two minor 27 arcs are congruent if and only if their corresponding 14. chords are congruent. So, The sum of the measures of the central angles of a circle with no interior points in common is 360. So, 2x = 245 x = 122.5 ANSWER: SOLUTION: In the same circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. Here, So, Therefore, 3x = 7x – 44 4x = 44 x = 11 ANSWER: 11 eSolutions Manual - Powered by Cognero Page 4 9-3 Arcs and Chords In , the radius is 14 and CD = 22. Find each In , the diameter is 18, LM = 12, and measure. Round to the nearest hundredth, if . Find each measure. Round to the necessary. nearest hundredth, if necessary. 16. CE 18. SOLUTION: SOLUTION: Radius is perpendicular to chord . So, by Diameter is perpendicular to chord . So, by Theorem 10.3, bisects . Therefore, CE = Theorem 10.3, bisects arc LKM. Therefore, ED. m(arc LK) = m(arc KM). By substitution, CE = (22) or 11 units. By substitution, m(arc LK) = (84) or 42. ANSWER: ANSWER: 11 42 17. EB 19. HP SOLUTION: SOLUTION: First find AE. Draw radius and create right Draw radius and create right triangle HLP. triangle ACE. The radius of the circle is 14, so AC = Diameter JK = 18 and the radius of a circle is half of 14. Since the radius is perpendicular to , the diameter, so HL = 9. Since the diameter is bisects by Theorem 10.3. So, CE = (22) or 11. perpendicular to , bisects by Theorem Use the Pythagorean Theorem to find AE. 10.3. So, LP = (12) or 6. Use the Pythagorean Theorem to find HP. By the Segment Addition Postulate, EB = AB – AE. Therefore, EB is 14 – 8.66 or about 5.34 units long. Therefore, HP is about 6.71 units long. ANSWER: ANSWER: 5.34 6.71 eSolutions Manual - Powered by Cognero Page 5 9-3 Arcs and Chords 20. SNOWBOARDING The snowboarding rail shown 21. ROADS The curved road at the right is part of , is an arc of a circle in which is part of the which has a radius of 88 feet. What is AB? Round to the nearest tenth. diameter. If is about 32% of a complete circle, what is ? SOLUTION: SOLUTION: The radius of the circle is 88 ft. So, CD = CB = 88. The sum of the measures of the central angles of a Also, CE = CD – ED = 88 – 15 = 73. circle with no interior points in common is 360. Use the Pythagorean Theorem to find EB, the length of a leg of the right triangle CEB. If a diameter (or radius) of a circle is perpendicular to The diameter containing is perpendicular to a chord, then it bisects the chord and its arc. So, chord . So, by Theorem 10.3, bisects arc bisects Therefore, ABC. Therefore, m(arc AB) = m(arc BC). By ANSWER: substitution, m(arc AB) = (115.2) or 57.6. 98.3 ft ANSWER: 22. ALGEBRA In , ,DF = 3x – 7, and FE 57.6 = x + 9. What is x? SOLUTION: In the same circle or in congruent circles, two chords are congruent if and only if they are equidistant from the center. Since , DF = EF. 3x – 7 = x + 9 2x = 16 x = 8 ANSWER: 8 eSolutions Manual - Powered by Cognero Page 6 9-3 Arcs and Chords 23. ALGEBRA In , LM = 16 and PN = 4x. What is 2. Draw radii and . (2 points determine a x? line.) 3. (All radii of a are .) 4. (Reflex. Prop. of ) 5. and are right . (Def. of ) 6. (All right are .) 7. (SAS) 8. (CPCTC) SOLUTION: 9. bisects .(Def. of bisect) In the same circle or in congruent circles, two chords 10. (CPCTC) are congruent if and only if they are equidistant from the center. Since SQ = SR, LM = PN. 11. (In the same circle, two arcs are 4x = 16 congruent if their corresponding central angles are x = 4 congruent.) 12. bisects . (Def. of bisect) ANSWER: 4 ANSWER: Given: PROOF Write a two-column proof. Prove: bisects and 24. Given: Prove: bisects and Proof: Statements (Reasons) 1. (Given) SOLUTION: 2. Draw radii and . (2 points determine a Given: line.) Prove: bisects and 3. (All radii of a are .) 4. (Reflex. Prop. of ) 5. and are right . (Def. of ) 6. (All right are .) 7. (SAS) 8. (CPCTC) 9. bisects .(Def. of bisect) 10. (CPCTC) Proof: 11. (In the same circle, two arcs are Statements (Reasons) congruent if their corresponding central angles are 1. (Given) congruent.) eSolutions Manual - Powered by Cognero Page 7 9-3 Arcs and Chords 12. bisects . (Def. of bisect) 26. two-column proof of Theorem 9.3 Given: PROOF Write the specified type of proof. Prove: 25. paragraph proof of Theorem 9.2, part 2 Given: Prove: SOLUTION: Proof: Statements (Reasons) SOLUTION: 1. (Given) Proof: 2. (All radii of a are .) Because all radii are congruent, 3. (Reflexive Prop.) 4. and are rt. (Definition of . You are given that , so by lines) SSS. Thus, by CPCTC. Since the 5. (HL) central angles have the same measure, their intercepted arcs have the same measure and are 6. , (CPCTC) therefore congruent. Thus, . 7. (If central are , intercepted arcs are .) ANSWER: ANSWER: Proof: Proof: Because all radii are congruent, Statements (Reasons) . You are given that , so by 1. (Given) SSS. Thus, by CPCTC. Since the 2. (All radii of a are .) central angles have the same measure, their 3. (Reflexive Prop.) intercepted arcs have the same measure and are 4. and are rt. (Definition of therefore congruent. Thus, . lines) 5. (HL) 6. , (CPCTC) 7. (If central are , intercepted arcs are .) eSolutions Manual - Powered by Cognero Page 8 9-3 Arcs and Chords 27. DESIGN Roberto is designing a logo for a friend’s 28. CONSTRUCT ARGUMENTS Write a two- coffee shop according to the design at the right, column proof of Theorem 9.4.
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