10-3 Arcs and Chords. ALGEBRA Find the value of x.
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1 ALGEBRA Find the value of x. 1. Arc ST is a minor arc, so m(arc ST) is equal to the measure of its related central angle or 93. and are congruent chords, so the corresponding arcs RS and ST are congruent. m(arc RS) = m(arc ST) and by substitution, x = Since HG = 4 and FG = 4, and are congruent chords and the corresponding arcs HG and FG are congruent. m(arc HG) = m(arc FG) = x Arc HG, arc GF, and arc FH are adjacent arcs that form the circle, so the sum of their measures is 360. esolutions Manual - Powered by Cognero Page 1
2 3. congruent. Since m(arc AB) = m(arc CD) = 127, arc AB arc CD and. In, JK = 10 and. Find each measure. Round to the nearest hundredth. 4. Radius is perpendicular to chord. So, by Theorem 10.3, bisects arc JKL. Therefore, m(arc JL) = m(arc LK). By substitution, m(arc JL) = or PQ 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 So, JQ = (10) or 5. Use the Pythagorean Theorem to find PQ. So, PQ is about 3.32 units long. esolutions Manual - Powered by Cognero Page 2
3 6. In, GH = 9, KL = 4x + 1. Find x. In the same circle or in congruent circles, two chords are congruent if and only if they are equidistant from the center. Since JS = JR, KL = GH. ALGEBRA Find the value of x. 7. congruent. Therefore, 5x = 105 x = congruent. So, The sum of the measures of the central angles of a circle with no interior points in common is 360. So, 2x = 254 x = 127 esolutions Manual - Powered by Cognero Page 3
4 congruent. Here, 3x + 5 = 26 x = 7 Therefore, congruent. Here, 5x = 3x + 54 x = 4 Therefore, 15. CCSS 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? 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 = esolutions Manual - Powered by Cognero Page 4
5 In, the radius is 14 and CD = 22. Find each measure. Round to the nearest hundredth, if necessary. 17. EB First find AE. Draw radius and create right triangle ACE. The radius of the circle is 14, so AC = 14. Since the radius is perpendicular to, bisects by Theorem So, CE = (22) or 11. Use the Pythagorean Theorem to find AE. By the Segment Addition Postulate, EB = AB - AE. Therefore, EB is or about 5.34 units long. esolutions Manual - Powered by Cognero Page 5
6 In, the diameter is 18, LM = 12, and. Find each measure. Round to the nearest hundredth, if necessary. 19. HP Draw radius and create right triangle HLP. Diameter JK = 18 and the radius of a circle is half of the diameter, so HL = 9. Since the diameter is perpendicular to, bisects by Theorem So, LP = (12) or 6. Use the Pythagorean Theorem to find HP. Therefore, HP is about 6.71 units long. 21. ROADS The curved road at the right is part of, which has a radius of 88 feet. What is AB? Round to the nearest tenth. The radius of the circle is 88 ft. So, CD = CB = 88. Also, CE = CD ED = = 73. 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 a chord, then it bisects the chord and its arc. So, Therefore, bisects esolutions Manual - Powered by Cognero Page 6
7 23. ALGEBRA In, LM = 16 and PN = 4x. What is x? In the same circle or in congruent circles, two chords are congruent if and only if they are equidistant from the center. Since SQ = SR, LM = PN. 4x = 16 x = 4 PROOF Write the specified type of proof. 25. paragraph proof of Theorem 10.2, part 2 Given: Prove: Proof: Because all radii are congruent,. You are given that, so by SSS. Thus, by CPCTC. Since the central angles have the same measure, their intercepted arcs have the same measure and are therefore congruent. Thus,. 27. DESIGN Roberto is designing a logo for a friend s coffee shop according to the design at the right, where each chord is equal in length. What is the measure of each arc and the length of each chord? The four chords are equal in length. So, the logo is a square inscribed in a circle. Each diagonal of the square is a diameter of the square and it is 3 ft long. The length of each side of a square of diagonal d units long is given by Therefore, the length of each chord is congruent. Here, all the four chords are equal in length and hence the corresponding arcs are equal in measure. Therefore, each arc esolutions Manual - Powered by Cognero Page 7
8 CCSS ARGUMENTS Write a two-column proof of the indicated part of Theorem In a circle, if two chords are equidistant from the center, then they are congruent. Given: Prove: Proof: Statements (Reasons) 1. (All radii of a are.) 2. (Given) 3. and are right. (Definition of lines) 4. (HL) 5. (CPCTC) 6. XG = YH (Definition of segments) 7. 2(XG) = 2(YH) (Multiplication Property of Equality) 8. bisects ; bisects. ( and are contained in radii. A radius to a chord bisects the chord.) 9. FG = 2(XG), JH = 2(YH) (Definition of segment bisector) 10. FG = JH (Substitution) 11. (Definition of segments) Find the value of x. 31. If a diameter (or radius) of a circle is perpendicular to a chord, then it bisects the chord and its arc. So, We have. Then, 9x = 2x x = 14 x = 2 esolutions Manual - Powered by Cognero Page 8
9 33. To find x we need to show that chords and are congruent. Use the values of the segments shown on the figure to find x. Therefore, the value of x is 5. esolutions Manual - Powered by Cognero Page 9
So, PQ is about 3.32 units long Arcs and Chords. ALGEBRA Find the value of x.
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