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Exercises 6.4 In Exercises 1 to 8, use the figure provided. 1. If mCD mAB, write an inequality that compares mZCQD and mZAQB. 2. If mCD QY > Qx, which chord has the greatest length? Which has the least length? Why? 17. Given circle O with radius OT, tangent AD, and line segments OA, OB, OC, and OD: a) Which line segment 18. 19. 20. 21. drawn from O has the smallest length? b) If m1 B R N 40°, m/2 = 50°, m3 = 45°, and m4 = 30°, which line segment from point O has the greatest length? a) If mRS mTV, write an inequality that compares mZ1 with m/2. b) If m1> m2, write an inequality that compares mRS with m TV. a) If MNPQ, write an inequality that compares the measures of minor arcs MN and PQ. b) If MNPQ, write an inequality that compares the measures of major arcs MPN and PMQ. a) If mXYmYZ write an inequality that compares the measures of inscribed angles 1 and 2. b) If m1

Exercises 6.4 In Exercises 1 to 8, use the figure provided. 1. If mCD mAB, write an inequality that compares mZCQD and mZAQB. 2. If mCD QY > Qx, which chord has the greatest length? Which has the least length? Why? 17. Given circle O with radius OT, tangent AD, and line segments OA, OB, OC, and OD: a) Which line segment 18. 19. 20. 21. drawn from O has the smallest length? b) If m1 B R N 40°, m/2 = 50°, m3 = 45°, and m4 = 30°, which line segment from point O has the greatest length? a) If mRS mTV, write an inequality that compares mZ1 with m/2. b) If m1> m2, write an inequality that compares mRS with m TV. a) If MNPQ, write an inequality that compares the measures of minor arcs MN and PQ. b) If MNPQ, write an inequality that compares the measures of major arcs MPN and PMQ. a) If mXYmYZ write an inequality that compares the measures of inscribed angles 1 and 2. b) If m1

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
Chapter6: Circles
Section6.1: Circles And Related Segments And Angles
Problem 18E: AB is the common chord of O and Q. If AB=12 and each circle has a radius of length 10, how long is...
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solve no 9,10,12,14,26

Exercises 6.4 In Exercises 1 to 8, use the figure provided. 1. If mCD mAB, write an inequality that compares mZCQD and mZAQB. 2. If mCD <mAB, write an inequality that compares CD and AB. M A B Exercises 1-8 3. If mCD mAB, write an inequality that compares QM and QN. 4. If mCD <mAB, write an inequality that compares mZA and mZC. 5. If mCQD < mZAQB, write an inequality that compares CD to AB. 6. If mCQD < mZAQB, write an inequality that compares QM to QN. Unless otherwise noted, all content on this page is Cengage Learning. 7. If mCD:mAB = 3:2, write an inequality that compares QM to QN. 8. If QN:QM = 5:6, write an inequality that compares MAB to mCD. 9. Construct a circle O and choose some point D on the circle. Now construct the tangent to circle O at point D. 10. Construct a circle P and choose three points R, S, and T on the circle. Construct the triangle that has its sides tangent to the circle at R, S, and T. 11. X, Y, and Z are on circle O such that mXY and mXZ 120°, m YZ = 130°, 110°. Suppose that triangle XYZ is drawn and that the triangle ABC is constructed with its sides tangent to circle O at X, Y, and Z. Are AXYZ and AABC similar triangles? Copyright 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 302 CHAPTER 6 CIRCLES 12. Construct the two tangent segments to circle P (not shown) from external point E. 13. Point V is in the exterior of circle Q (not shown) such that VQ is equal in length to the diameter of circle Q. Construct the two tangents to circle Q from point V. Then determine the measure of the angle that has vertex V and has the tan- gents as sides. 14. Given circle P and points R-P-T such that R and T are in the exterior of circle P, suppose that tangents are constructed from R and T to form a quadrilateral (as shown). Identify the type of quadrilateral formed a) when RP PT. b) when RP = PT. 15. Given parallel chords AB, CD, EF, and GH in circle O, which chord has the greatest length? Which has the least length? Why? 16. Given chords MN, RS, and TV in OQ such that QZ > QY > Qx, which chord has the greatest length? Which has the least length? Why? 17. Given circle O with radius OT, tangent AD, and line segments OA, OB, OC, and OD: a) Which line segment 18. 19. 20. 21. drawn from O has the smallest length? b) If m1 B R N 40°, m/2 = 50°, m3 = 45°, and m4 = 30°, which line segment from point O has the greatest length? a) If mRS mTV, write an inequality that compares mZ1 with m/2. b) If m1> m2, write an inequality that compares mRS with m TV. a) If MNPQ, write an inequality that compares the measures of minor arcs MN and PQ. b) If MNPQ, write an inequality that compares the measures of major arcs MPN and PMQ. a) If mXYmYZ write an inequality that compares the measures of inscribed angles 1 and 2. b) If m1 <m/2, write an inequality that compares the measures of XY and YZ D R S Y N Quadrilateral ABCD is inscribed in circle P (not shown). If ZA is an acute angle, what type of angle is <C? 22. Quadrilateral RSTV is inscribed in circle Q (not shown). If arcs RS, ST, and TV are all congruent, what type of quadrilateral is RSTV? 23. In circle O, points A, B, and C are on the circle such that mAB = 60° and mBC 40°. a) How are mZAOB and mZBOC related? b) How are AB and BC related? 24. In OO, AB = 6cm and BC 4 cm. a) How are mZAOB and mZBOC related? b) How are mAB and mBC related? Exercises 23-25 25. In 00, mZAOB = 70° and mZBOC = 30°. See the figure above. a) How are mAB and mBC related? b) How are AB and BC related? 26. Triangle ABC is inscribed in circle O; AB = 5, BC = 6, and AC = 7. a) Which is the largest minor arc of OO: AB, BC, or AC? b) Which side of the triangle is nearest point O? 27. Given circle O with mBC = 120° 28. 28 and mAC 130°: Exercises 26-29 a) Which angle of triangle ABC is smallest? b) Which side of triangle ABC is nearest point O? Given that mAC:mBC:mAB = 4:3:2 in circle O: a) Which arc is largest? b) Which chord is longest? 29. Given that mZA:m/B:mC = 2:4:3 in circle O: a) Which angle is largest? b) Which chord is longest? (NOTE: See the figure for Exercise 26.) 30. Circle O has a diameter of length 20 cm. Chord AB has length 12 cm, and chord CD has length 10 cm. How much closer is AB than CD to point O? 31. Circle P has a radius of length 8 in. Points A, B, C, and D lie on circle P in such a way that mZAPB = 90° and mZCPD = 60°. How much closer to point P is chord AB than chord CD? 32. A tangent ET is constructed to circle Q from external point E. Which angle and which side of triangle QTE are largest? Which angle and which side are smallest? 33. Two congruent circles, 0 and OP, do not intersect. Construct a common external tangent for 0 and OP. 34. Explain why the following statement is incorrect: "In a circle (or in congruent circles) containing two unequal chords, the longer chord corresponds to the greater major arc." 35. Prove: In a circle containing two unequal arcs, the larger arc corresponds to the larger central angle. 36. Prove: In a circle containing two unequal chords, the longer chord corresponds to the larger central angle. (HINT: You may use any theorems stated in this section.) Unless otherwise noted, all content on this page is Cengage Learning. Copyright 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. *37. In 0, chord AB || chord CD. Radius OE is perpendicular to AB and CD at points M and N, respectively. If OE = 13, AB 24, and CD = 10, then the distance from O to CD is greater than the distance from O to AB. Determine how much farther chord CD is from center O than chord AB is from center O; that is, find MN. Perspective on History 303 *38. In OP, whose radius has length 8 in., mAB = mBC = 60°. Because mAC 120°, chord AC is longer than either of the congruent chords AB and BC. Determine how much longer AC is than AB; that is, find the exact value and the approximate value of ACAB. N E B 39. Construct two externally tangent circles where the radius length of one circle is twice the radius length of the other circle.
Transcribed Image Text:Exercises 6.4 In Exercises 1 to 8, use the figure provided. 1. If mCD mAB, write an inequality that compares mZCQD and mZAQB. 2. If mCD <mAB, write an inequality that compares CD and AB. M A B Exercises 1-8 3. If mCD mAB, write an inequality that compares QM and QN. 4. If mCD <mAB, write an inequality that compares mZA and mZC. 5. If mCQD < mZAQB, write an inequality that compares CD to AB. 6. If mCQD < mZAQB, write an inequality that compares QM to QN. Unless otherwise noted, all content on this page is Cengage Learning. 7. If mCD:mAB = 3:2, write an inequality that compares QM to QN. 8. If QN:QM = 5:6, write an inequality that compares MAB to mCD. 9. Construct a circle O and choose some point D on the circle. Now construct the tangent to circle O at point D. 10. Construct a circle P and choose three points R, S, and T on the circle. Construct the triangle that has its sides tangent to the circle at R, S, and T. 11. X, Y, and Z are on circle O such that mXY and mXZ 120°, m YZ = 130°, 110°. Suppose that triangle XYZ is drawn and that the triangle ABC is constructed with its sides tangent to circle O at X, Y, and Z. Are AXYZ and AABC similar triangles? Copyright 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 302 CHAPTER 6 CIRCLES 12. Construct the two tangent segments to circle P (not shown) from external point E. 13. Point V is in the exterior of circle Q (not shown) such that VQ is equal in length to the diameter of circle Q. Construct the two tangents to circle Q from point V. Then determine the measure of the angle that has vertex V and has the tan- gents as sides. 14. Given circle P and points R-P-T such that R and T are in the exterior of circle P, suppose that tangents are constructed from R and T to form a quadrilateral (as shown). Identify the type of quadrilateral formed a) when RP PT. b) when RP = PT. 15. Given parallel chords AB, CD, EF, and GH in circle O, which chord has the greatest length? Which has the least length? Why? 16. Given chords MN, RS, and TV in OQ such that QZ > QY > Qx, which chord has the greatest length? Which has the least length? Why? 17. Given circle O with radius OT, tangent AD, and line segments OA, OB, OC, and OD: a) Which line segment 18. 19. 20. 21. drawn from O has the smallest length? b) If m1 B R N 40°, m/2 = 50°, m3 = 45°, and m4 = 30°, which line segment from point O has the greatest length? a) If mRS mTV, write an inequality that compares mZ1 with m/2. b) If m1> m2, write an inequality that compares mRS with m TV. a) If MNPQ, write an inequality that compares the measures of minor arcs MN and PQ. b) If MNPQ, write an inequality that compares the measures of major arcs MPN and PMQ. a) If mXYmYZ write an inequality that compares the measures of inscribed angles 1 and 2. b) If m1 <m/2, write an inequality that compares the measures of XY and YZ D R S Y N Quadrilateral ABCD is inscribed in circle P (not shown). If ZA is an acute angle, what type of angle is <C? 22. Quadrilateral RSTV is inscribed in circle Q (not shown). If arcs RS, ST, and TV are all congruent, what type of quadrilateral is RSTV? 23. In circle O, points A, B, and C are on the circle such that mAB = 60° and mBC 40°. a) How are mZAOB and mZBOC related? b) How are AB and BC related? 24. In OO, AB = 6cm and BC 4 cm. a) How are mZAOB and mZBOC related? b) How are mAB and mBC related? Exercises 23-25 25. In 00, mZAOB = 70° and mZBOC = 30°. See the figure above. a) How are mAB and mBC related? b) How are AB and BC related? 26. Triangle ABC is inscribed in circle O; AB = 5, BC = 6, and AC = 7. a) Which is the largest minor arc of OO: AB, BC, or AC? b) Which side of the triangle is nearest point O? 27. Given circle O with mBC = 120° 28. 28 and mAC 130°: Exercises 26-29 a) Which angle of triangle ABC is smallest? b) Which side of triangle ABC is nearest point O? Given that mAC:mBC:mAB = 4:3:2 in circle O: a) Which arc is largest? b) Which chord is longest? 29. Given that mZA:m/B:mC = 2:4:3 in circle O: a) Which angle is largest? b) Which chord is longest? (NOTE: See the figure for Exercise 26.) 30. Circle O has a diameter of length 20 cm. Chord AB has length 12 cm, and chord CD has length 10 cm. How much closer is AB than CD to point O? 31. Circle P has a radius of length 8 in. Points A, B, C, and D lie on circle P in such a way that mZAPB = 90° and mZCPD = 60°. How much closer to point P is chord AB than chord CD? 32. A tangent ET is constructed to circle Q from external point E. Which angle and which side of triangle QTE are largest? Which angle and which side are smallest? 33. Two congruent circles, 0 and OP, do not intersect. Construct a common external tangent for 0 and OP. 34. Explain why the following statement is incorrect: "In a circle (or in congruent circles) containing two unequal chords, the longer chord corresponds to the greater major arc." 35. Prove: In a circle containing two unequal arcs, the larger arc corresponds to the larger central angle. 36. Prove: In a circle containing two unequal chords, the longer chord corresponds to the larger central angle. (HINT: You may use any theorems stated in this section.) Unless otherwise noted, all content on this page is Cengage Learning. Copyright 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. *37. In 0, chord AB || chord CD. Radius OE is perpendicular to AB and CD at points M and N, respectively. If OE = 13, AB 24, and CD = 10, then the distance from O to CD is greater than the distance from O to AB. Determine how much farther chord CD is from center O than chord AB is from center O; that is, find MN. Perspective on History 303 *38. In OP, whose radius has length 8 in., mAB = mBC = 60°. Because mAC 120°, chord AC is longer than either of the congruent chords AB and BC. Determine how much longer AC is than AB; that is, find the exact value and the approximate value of ACAB. N E B 39. Construct two externally tangent circles where the radius length of one circle is twice the radius length of the other circle.
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