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Answers for the lesson “Find Segment Lengths in Circles” LESSON 10.6 Skill Practice 21. Statements (Reasons) 1. external segment 2. A tangent segment intersects the circle in only one point while the secant segment intersects the circle in two points. 3. 5 4. 23 5. 4 6. 4 7. 6 8. 5 9. 12 10. 36 11. 4 12. The wrong segment lengths are Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. being multiplied together; FD + CF 5 AF + BF, 4 + CF 5 3 + 8, CF 5 6, CD 1 DF 5 CF, CD 1 4 5 6, CD 5 2. 13. 5 16. D 14. 30 15. 1 17. 18 18. 17.7 } 19. 2Ï 10 Problem Solving 20. about 108 ft 1. Two intersecting chords in the same circle. (Given) 2. Draw } AC and } BD. (Two points determine a line.) 3. ACD > ABD, CAB > CDB (If two inscribed angles of a circle intercept the same arc, then the angles are congruent.) 4. nAEC , nDEB (AA Similarity Postulate) EA EC 5. } 5} EB ED (If two triangles are similar, the ratios of corresponding sides are equal.) 6. EA + EB 5 EC + ED (Cross Products Property) AD + (AD 1 DE) 22. BC 5 }} 2 AB AB Geometry Answer Transparencies for Checking Homework 316 23. AC , which implies mD 5 }2mC 1 A x E y r C r P r D Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. } is a tangent segment to Given: EA } is a secant segment of circle P, ED } is a diameter of circle P, and CD circle P. Prove: EA2 5 EC ? ED. } > AP }, By Theorem 10.1, EA so nEAP is a right triangle. By the Pythagorean Theorem, (y 1 r)2 = x2 1 r2. So, y2 1 2yr 1 r2 5 x2 1 r2. By the Subtraction Property of Equality, y2 1 2yr 5 x2. Factoring this, y(y 1 2r) 5 x2, or EC + ED 5 EA2. 24. 4 cm/sec; by the Segments of Chords Theorem, 6 + 8 5 4 + CN, so CN 5 12 cm. It takes 3 sec for sparkles to move the 6 cm from C to D, so the sparkles need to travel 12 cm in 3 sec, or 4 cm/sec. 25. Given: } EB and } ED are secant segments. Draw } AD and } BC. Using the Measure of an Inscribed Angle Theorem, mB 5 } mC AC and 2 1 B > D. Using the Reflexive Property of Angle Congruence, E > E. Using the AA Similarity Postulate, nBCE , nDAE. Using corresponding sides of similar triangles are proportional, EA EC ED EB } 5 }. Cross multiplying gives EA + EB 5 EC + ED. 26. Given: A secant segment and a tangent segment sharing an endpoint outside of a circle. Draw } AC and } AD. ADC is inscribed, therefore mADC 5 }2 mC AC . 1 CAE is formed by a secant and a tangent, therefore mCAE 5 }2 mC AC . This 1 implies ADC > CAE. E > E by the Reflexive Property of Angle Congruence, therefore n AEC , n DEA using the AA Similarity Postulate. Using corresponding sides of similar triangles are proportional, EA EC ED EA } 5 }. Cross multiplying gives EA2 5 EC + ED. Geometry Answer Transparencies for Checking Homework 317 27. a. 608 b. Using the Vertical Angles Theorem, ACB > FCE. Since mCAB 5 608 and mEFD 5 608, then CAB > EFD. Using the AA Similarity Postulate, n ABC , n FEC. y x 1 10 x 1 10 c. } 5 }; y 5 } 6 2 3 d. y 2 5 x(x 1 16) e. 2, 6 2 CE f. Since } 5 }, let CE 5 2x 1 CB Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. and CB 5 x. Using Theorem 10.14, 2x2 5 60 which } implies x 5 Ï30 which } implies CE 5 2Ï 30 . } 25Ï 29 28. } 29 Geometry Answer Transparencies for Checking Homework 318