Skills Practice Skills Practice for Lesson 11.1

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1 Skills Practice Skills Practice for Lesson.1 Name ate Riding a Ferris Wheel Introduction to ircles Vocabulary Identify an instance of each term in the diagram. 1. circle X T 2. center of the circle H I 3. chord of the circle U V 4. diameter M P N 5. secant of the circle 6. tangent of the circle 7. point of tangency 8. central angle 9. inscribed angle 10. arc. major arc hapter Skills Practice 807

2 12. minor arc Problem Set Identify the indicated part of each circle. Explain your answer GE G F E is a radius. It is a line segment that connects the center to a point on the circle, NP H I N M J K P F E 808 hapter Skills Practice

3 Name ate 7. JH 8. MN H L J M I K N P 9. SQR 10. TU S R Q V W T U Use circle P to answer each question.. Which point(s) are located in the interior of the circle? Point P is in the interior of circle P. 12. Which point(s) are located on the circle? 13. Which point(s) are located in the exterior of the circle? M P T R Y 14. Why isn t MR a diameter? 15. How are MR and R alike? hapter Skills Practice 809

4 16. How are MR and R different? Identify each angle as an inscribed angle or a central angle. 17. URE R ngle URE is an inscribed angle. 18. ZM 19. KM Z K M 20. ZKU E U 21. MU 22. RK lassify each arc as a major arc, a minor arc, or a semicircle E 25. FHI is a minor arc. F 26. JML E F G H J M K I L 810 hapter Skills Practice

5 Name ate 27. NPQ 28. TRS N P R S Q T U raw the part of a circle that is described. 29. raw chord. 30. raw radius E. 31. raw secant GH. 32. raw a tangent at point J. J 33. Label the point of tangency. 34. Label center. hapter Skills Practice 8

6 35. raw inscribed angle FG. 36. raw central angle HI. 37. raw major arc KNM. 38. raw minor arc RQ. 812 hapter Skills Practice

7 Skills Practice Skills Practice for Lesson.2 Name ate Holding the Wheel entral ngles, Inscribed ngles, and Intercepted rcs Vocabulary efine each term in your own words. 1. degree measure of a minor arc 2. adjacent arcs 3. rc ddition Postulate 4. intercepted arc 5. Inscribed ngle Theorem 6. Parallel Lines-ongruent rcs Theorem hapter Skills Practice 813

8 Problem Set etermine the measure of each minor arc The measure of is 90º. 3. EF 4. GH 45 E 135 H F G 5. IJ 6. KL J 120 I K 85 L 814 hapter Skills Practice

9 Name ate etermine the measure of each central angle. 7. m XYZ 8. m GT X 30 P 0 R Y Z G T m XYZ m LKJ 10. m FMR 31 R L 30 K M 98 J N 72 F M. m KWS 12. m VIQ V K W 22 N S T 85 I Q hapter Skills Practice 815

10 etermine the measure of each inscribed angle. 13. m XYZ 14. m MTU 150 X Z T Q 82 U Y M m XYZ m KLS 16. m V V K L 2 S m QR 18. m SGI Q 155 R G S 28 I 816 hapter Skills Practice

11 Name ate etermine the measure of each intercepted arc. 19. m KM K 20. m IU L 36 M 54 m KM 108 L I U 21. m QW 22. m TV Q W T V 23. m ME 24. m S R 52 E M 90 P S hapter Skills Practice 817

12 alculate the measure of each angle. 25. The measure of is 62º. What is the measure of? m 1 2 (m ) 62º 31º The measure of is 98º. What is the measure of E? E 27. The measure of EG is 128º. What is the measure of EFG? E G F 818 hapter Skills Practice

13 Name ate 28. The measure of GH is 74º. What is the measure of GIH? G H I 29. The measure of JK is 168º. What is the measure of JIK? I J K 30. The measure of KL is 148º. What is the measure of KML? K L M hapter Skills Practice 819

14 Use the given information to answer each question. 31. In circle, m XZ 86º. What is m WY? X W Y m WY In circle, m WX 102º. What is m YZ? Z X W Y Z 33. In circle, m WZ 65º and m XZ 38º. What is m WX? Z X W 820 hapter Skills Practice

15 Name ate 34. In circle, m WX 105º. What is m WYX? X W Y 35. In circle, m WY 83º. What is m XZ? X W Y Z 36. In circle, m WYX 50º and m XYZ 30º. What is m WXZ? X W Y Z hapter Skills Practice 821

16 822 hapter Skills Practice

17 Skills Practice Skills Practice for Lesson.3 Name ate Manhole overs Measuring ngles Inside and utside of ircles Vocabulary Identify the similarities and differences between the terms. 1. Interior ngles of a ircle Theorem and Exterior ngles of a ircle Theorem Problem Set List the two intercepted arcs for the given angle. 1. E 2. MRN M, E 3. VYU 4. KNL U Q J P R K N X Y V N L W M hapter Skills Practice 823

18 5. EIF 6. SUT E G R Q U S F I H T Write an expression for the measure of the given angle. 7. m RPM 8. m R P M N E Q m RPM 1 2 ( m RM m QN ) 9. m JNK 10. m UWV L M N J K U V W Y X 824 hapter Skills Practice

19 Name ate. m SWT 12. m HJI S W T F V G U H J I List the intercepted arc(s) for the given angle. 13. QMR 14. RSU Q R S T M N P R P U NP, QR 15. FEH 16. ZW E G F H I Z X Y W hapter Skills Practice 825

20 17. E 18. JLM J Z Z E K M L Write an expression for the measure of the given angle. 19. m 20. m UXY E W X Y U Z m 1 2 ( m E m ) 21. m SRT 22. m FJG R S U V T F H I G J 826 hapter Skills Practice

21 Name ate 23. m EG 24. m LPN E F H L M G N p Q reate a two-column proof to prove each statement. 25. Given: hords E and intersect at point. Prove: m 1 2 ( m m E ) E Statements Reasons 1. hords E and intersect at 1. Given point. 2. raw chord 2. onstruction 3. m m m 3. Exterior ngle Theorem 4. m 1 2 m E 4. Inscribed ngle Theorem 5. m 1 2 m 5. Inscribed ngle Theorem 6. m 1 2 m E 1 2 m 6. Substitution 7. m 1 m 2 m E ) 7. istributive Property hapter Skills Practice 827

22 26. Given: Secant QT and tangent SR intersect at point S. Prove: m QSR 1 2 ( m QR m RT ) Q P R T S 828 hapter Skills Practice

23 Name ate 27. Given: Tangents VY and XY intersect at point Y. Prove: m Y 1 2 ( m VWX m VX ) V Y W X hapter Skills Practice 829

24 28. Given: hords FI and GH intersect at point J. Prove: m FJH 1 2 ( m FH m GI ) F G J H I 830 hapter Skills Practice

25 Name ate 29. Given: Secant JL and tangent NL intersect at point L. Prove: m L 1 2 ( m JM m KM ) K L J M N hapter Skills Practice 831

26 30. Given: Tangents SX and UX intersect at point X. Prove: m X 1 2 ( m VTW m VW ) T S U V W X 832 hapter Skills Practice

27 Name ate Use the diagram shown to determine the measure of each angle or arc. 31. etermine m FI. m K 20 m GJ etermine m KLJ. m KM 120 m JN 100 F G J K K J L I M N m FI etermine m X. 34. etermine m WYX. m VW 50 m TU 85 m WUY 300 R S W T U U V X W X Y Z hapter Skills Practice 833

28 35. etermine m RS. m UV 30 m RTS etermine m. m ZX 150 m 30 R S Z T X U V 834 hapter Skills Practice

29 Skills Practice Skills Practice for Lesson.4 Name ate olor Theory hords Vocabulary Match each definition with its corresponding term. 1. iameter-hord Theorem a. If two chords of the same circle or congruent circles are congruent, then their corresponding arcs are congruent. 2. Equidistant hord Theorem b. The segments formed on a chord when two chords of a circle intersect. 3. Equidistant hord c. If two chords of the same circle or congruent onverse Theorem circles are congruent, then they are equidistant from the center of the circle. 4. ongruent hord-ongruent d. If two arcs of the same circle or congruent rc Theorem circles are congruent, then their corresponding chords are congruent. 5. ongruent hord-ongruent rc e. If two chords of the same circle or congruent onverse Theorem circles are equidistant from the center of the circle, then the chords are congruent. 6. segments of a chord f. If two chords of a circle intersect, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments in the second chord. 7. Segment-hord Theorem g. If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and bisects the arc determined by the chord. hapter l Skills Practice 835

30 Problem Set Use the given information to answer each question. Explain your answer. 1. If diameter bisects, what is the angle of intersection? The angle of intersection is 90º because diameters that bisect chords are perpendicular bisectors. 2. If diameter FH intersects EG at a right angle, how does the length of EI compare to the length of IG? F E I G H 3. How does the measure of KL and LM compare? L N K J M 836 hapter Skills Practice

31 Name ate 4. If KP LN, how does the length of Q compare to the length of R? K L J Q R M P N 5. If Y Z, what is the relationship between TU and XV? S U Y T Z V X W 6. If G H and diameter EJ is perpendicular to both, what is the relationship between GF and HK? I J E G H F K hapter Skills Practice 837

32 etermine each measurement. 7. If is a diameter, what is the length of E? 5 cm E E E 5 cm 8. If the length of is 13 millimeters, what is the length of? 9. If the length of is 24 centimeters, what is the length of? 10. If the length of F is 32 inches, what is the length of H? F G E H 838 hapter Skills Practice

33 Name ate. If the measure of 155º, what is the measure of? 12. If segment is a diameter, what is the measure of E? E ompare each measurement. Explain your answer. 13. If E FG, how does the measure of E and FG compare? E F G The measure of E is equal to the measure of FG because congruent segments on the same circle create congruent chords. hapter Skills Practice 839

34 14. If KM JL, how does the measure of JL and KM compare? K J L M 15. If QR PS, how does the measure of QPR and PRS compare? Q P R S 16. If EG EH, how does the measure of EG and H compare? G E F H 840 hapter Skills Practice

35 Name ate 17. If, what is the relationship between and? 18. If EH GF, what is the relationship between EH and FG? E H F G etermine each measurement. 19. If is congruent to, what is the length of? 17 cm 17 cm hapter Skills Practice 841

36 20. If EF is congruent to GH, what is the length of EF? F G E H 9 in. 21. If XU YV, what is the measure of XU? X Y 75 U V 22. If, what is the measure of? If the length of is 24 centimeters, what is the length of? 842 hapter Skills Practice

37 Name ate 24. If the length of EF is 14 millimeters, what is the length of HG? E F H G Use the given circles to answer each question. 25. Name the segments of chord KL. KN and NL K J 26. Name the segments of chord MJ. N 27. Name the segments of chord RU. M S L 28. Name the segments of chord SV. R T U V 29. Name the segments of chord. 30. Name the segments of chord. E hapter Skills Practice 843

38 Use each diagram and the Segment hord Theorem to write an equation involving the segments of the chords. 31. F 32. P G L Q N H J R G GJ FG GH 33. T V 34. S K H V G 35. I 36. X J E Y U L V 844 hapter Skills Practice

39 Skills Practice Skills Practice for Lesson.5 Name ate Solar Eclipses Tangents and Secants Vocabulary Write the term from the box that best completes each statement. external secant segment Secant-Segment Theorem tangent segment secant segment Secant-Tangent Theorem Tangent-Segment Theorem 1. (n) is the segment that is formed from an exterior point of a circle to the point of tangency. 2. The states that if two tangent segments are drawn from the same point on the exterior of the circle, then the tangent segments are congruent. 3. When two secants intersect in the exterior of a circle, the segment that begins at the point of intersection, continues through the circle, and ends on the other side of the circle is called a(n). 4. When two secants intersect in the exterior of a circle, the segment that begins at the point of intersection and ends where the secant enters the circle is called a(n). 5. The states that if two secants intersect in the exterior of a circle, then the product of the lengths of the secant segment and its external secant segment is equal to the product of the lengths of the second secant segment and its external secant segment. 6. The states that if a tangent and a secant intersect in the exterior of a circle, then the product of the lengths of the secant segment and its external secant segment is equal to the square of the length of the tangent segment. hapter Skills Practice 845

40 Problem Set alculate the measure of each angle. Explain your reasoning. 1. If is a radius, what is the measure of? tangent line and the radius that ends at the point of tangency are perpendicular, so m 90º. 2. If is a radius, what is the measure of? 3. If Y is a radius, what is the measure of XY? X Y 846 hapter Skills Practice

41 Name ate 4. If RS is a tangent segment and S is a radius, what is the measure of RS? R 35 S 5. If UT is a tangent segment and U is a radius, what is the measure of TU? 23 T U hapter Skills Practice 847

42 6. If V W is a tangent segment and V is a radius, what is the measure of VW? V 72 W Write a statement to show the congruent segments. 7. and are tangent to circle. 8. XZ and ZW are tangent to circle. X Z W 9. RS and RT are tangent to circle. 10. MP and NP are tangent to circle. R S M T P N 848 hapter Skills Practice

43 Name ate. E and FE are tangent to circle. 12. GH and GI are tangent to circle. H E G F I alculate the measure of each angle. Explain your reasoning. 13. If EF and GF are tangent segments, what is the measure of EGF? F 64 E G ecause EF and GF are tangent segments drawn from point F, they must be congruent. This fact means that EFG is an isosceles triangle. m EGF (180º 64º) 2 6º 2 58º The measure of EGF is 58º. hapter Skills Practice 849

44 14. If HI and JI are tangent segments, what is the measure of HJI? H 48 I J 15. If KM and LM are tangent segments, what is the measure of KML? K L 63 M 850 hapter Skills Practice

45 Name ate 16. If NP and QP are tangent segments, what is the measure of NPQ? N P 71 Q 17. If F and VF are tangent segments, what is the measure of VF? V 22 F hapter Skills Practice 851

46 18. If RT and MT are tangent segments, what is the measure of RTM? R T 33 M Name two secant segments and two external secant segments for circle P Q E R T S 21. Secant segments: PT and External secant segments: U X V QT RT and ST W 22. L M N P Y R 852 hapter Skills Practice

47 Name ate 23. F 24. H J J N G I K M L Use each diagram and the Secant Segment Theorem to write an equation involving the secant segments. 25. S R T U V 26. M N Q S R 27. RV TV SV UV E 28. I H J K L hapter Skills Practice 853

48 V F W E X Y Z Name a tangent segment, a secant segment, and an external secant segment for circle. 31. S T 32. I R U H J K 33. Tangent segment: TU Secant segment: RT External secant segment: F G ST E 34. P R Q S 854 hapter Skills Practice

49 Name ate W E H X y F G Z Use each diagram and the Secant Tangent Theorem to write an equation involving the secant and tangent segments. 37. Q W 38. S E M V R 39. (EM) 2 QM WM G Z X 40. X T I hapter Skills Practice 855

50 41. U 42. M l V E Q W 856 hapter Skills Practice

51 Skills Practice Skills Practice for Lesson.6 Name ate Replacement for a arpenter s Square Inscribed and ircumscribed Triangles and Quadrilaterals Vocabulary nswer each question. 1. How are inscribed polygons and circumscribed polygons different? 2. escribe how you can use the Inscribed Right Triangle-iameter Theorem to show an inscribed triangle is a right triangle. 3. What does the onverse of the Inscribed Right Triangle-iameter Theorem help to show in a circle? 4. What information about a quadrilateral inscribed in a circle does the Inscribed Quadrilateral-pposite ngles Theorem give? hapter Skills Practice 857

52 Problem Set raw a triangle inscribed in the circle through the three points. Then determine if the triangle is a right triangle No. The triangle is not a right triangle. None of the sides of the triangle is a diameter of the circle hapter Skills Practice

53 Name ate raw a triangle inscribed in the circle through the given points. Then determine the measure of the indicated angle. 9. In, m 55. etermine m. m In, m 38. etermine m. hapter Skills Practice 859

54 . In, m 62. etermine m. 12. In, m 26. etermine m. 13. In, m 49. etermine m. 14. In, m 51. etermine m. 860 hapter Skills Practice

55 Name ate raw a quadrilateral inscribed in the circle through the given four points. Then determine the measure of the indicated angle. 15. In quadrilateral, m 81. etermine m. m In quadrilateral, m 75. etermine m. 17. In quadrilateral, m 2. etermine m. hapter Skills Practice 861

56 18. In quadrilateral, m 93. etermine m. 19. In quadrilateral, m 72. etermine m. 20. In quadrilateral, m 101. etermine m. 862 hapter Skills Practice

57 Name ate onstruct a circle inscribed in each polygon hapter Skills Practice 863

58 hapter Skills Practice

59 Name ate 26. reate a two-column proof to prove each statement. 27. Given: Inscribed in circle, m 40, and m 140 Prove: is a diameter of circle Statements Reasons 1. m 40, m Given 2. m m m rc ddition Postulate m Substitution 4. m Subtraction Property of Equality 5. m 1 2 m 5. efinition of inscribed angle 6. m Substitution 7. is a right triangle with right angle. 7. efinition of right triangle 8. is the diameter of circle. 8. onverse of Inscribed Right Triangle-iameter Theorem hapter Skills Practice 865

60 28. Given: Inscribed RST in circle with diameter RS, and m RT 60 Prove: m ST 120 R S 60 T 866 hapter Skills Practice

61 Name ate 29. Given: Inscribed quadrilateral in circle, m 50, and m 90 Prove: m hapter Skills Practice 867

62 30. Given: Inscribed quadrilateral MNPQ in circle, m MQ 75, and m NMQ 120 Prove: m MN M 120 N Q P 868 hapter Skills Practice

63 Name ate 31. Given: m 50, m 90, and m 90 Prove: m 90 hapter Skills Practice 869

64 32. Given: and are supplementary angles Prove: m m 870 hapter Skills Practice

65 Skills Practice Skills Practice for Lesson.7 Name ate Gears rc Length Vocabulary efine the key term in your own words. 1. arc length Problem Set alculate the ratio of the length of each arc to the circle s circumference. 1. The measure of is 40º. 40º 360º 1 9 The arc is 1 of the circle s 9 circumference. 2. The measure of is 90º. 3. The measure of EF is 120º. 4. The measure of GH is 150º. 5. The measure of IJ is 105º. 6. The measure of KL is 75º. hapter Skills Practice 871

66 Write an expression that you can use to calculate the length of RS. You do not need to simplify the expression. 7. R R 0 15 in. S 13 cm S (15) R R 90 S 17 cm cm S. R 12. R 63 ft S yd alculate each arc length. Write your answer in terms of. 13. If the measure of is 45º and the radius is 12 meters, what is the arc length of? S (12) 24 Fraction of : 45º 1 360º 8 rc length of : 1 (24 ) 3 8 The arc length of is 3 meters. 872 hapter Skills Practice

67 Name ate 14. If the measure of is 120º and the radius is 15 centimeters, what is the arc length of? 15. If the measure of EF is 60º and the radius is 8 inches, what is the arc length of EF? 16. If the measure of GH is 30º and the radius is 6 meters, what is the arc length of GH? 17. If the measure of IJ is 80º and the diameter is 10 centimeters, what is the arc length of IJ? 18. If the measure of KL is 15º and the diameter is 18 feet, what is the arc length of KL? hapter Skills Practice 873

68 19. If the measure of MN is 75º and the diameter is 20 millimeters, what is the arc length of MN? 20. If the measure of P is 165º and the diameter is 21 centimeters, what is the arc length of P? alculate each arc length. Write your answer in terms of. 21. If the measure of is 135º, what is the arc length of? 16 cm 2 (16) 32 Fraction of : 135º 360º 3 8 rc length of : 3 (32 ) The arc length of is 12 cm. 874 hapter Skills Practice

69 Name ate 22. If the measure of is 45º, what is the arc length of? 18 cm 23. If the measure of EF is 90º, what is the arc length of EF? E 30 in. F hapter Skills Practice 875

70 24. If the measure of GH is 120º, what is the arc length of GH? G 27 in. H 25. If the length of the radius is 4 centimeters, what is the arc length of IJ? I 160 J 876 hapter Skills Practice

71 Name ate 26. If the length of the radius is 7 centimeters, what is the arc length of KL? K L If the length of the radius is centimeters, what is the arc length of MN? M 100 N hapter Skills Practice 877

72 28. If the length of the radius is 17 centimeters, what is the arc length of P? 75 P Use the given information to answer each question. Where necessary, use 3.14 to approximate. 29. etermine the perimeter of the shaded region. 30 mm 60 rc length: 60 2(3.14)(30) 31.4 mm 360 Perimeter of shaded region: mm 878 hapter Skills Practice

73 Name ate 30. etermine the perimeter of the figure below. 15 cm 30 cm semicircular cut was taken from the rectangle shown. etermine the perimeter of the shaded region. 80 ft 60 ft 32. circle has a circumference of 81.2 inches. What is the radius of the circle? hapter Skills Practice 879

74 33. ella used a tape measure and found the circumference of a flagpole to be 6.2 inches. What is the radius of the flagpole? 34. arla used a string and a tape measure and found the circumference of a circular cup to be inches. What is the radius of the cup? 880 hapter Skills Practice

75 Skills Practice Skills Practice for Lesson.8 Name ate Playing arts Sectors and Segments of a ircles Vocabulary raw an example of each term. 1. concentric circles 2. sector of a circle 3. segment of a circle Problem Set alculate the area of each circle. Write your answer in terms of. 1. What is the area of a circle whose radius is 5 centimeters? (5 2 ) 25 The area of the circle is 25 cm What is the area of a circle whose radius is 8 millimeters? hapter Skills Practice 881

76 3. What is the area of a circle whose radius is 12 feet? 4. What is the area of a circle whose radius is 18 centimeters? 5. What is the area of a circle whose diameter is 22 inches? 6. What is the area of a circle whose diameter is 28 meters? 7. What is the area of a circle whose diameter is 15 inches? 8. What is the area of a circle whose diameter is 31 yards? 882 hapter Skills Practice

77 Name ate alculate the area of each sector. Write your answer in terms of. 9. If the radius of the circle is 9 centimeters, what is the area of sector? 120 Total area of the circle (9 2 ) 81 cm 2 Sector s fraction of the circle 120º 1 360º 3 rea of sector 1 (81 ) 27 3 The area of sector is 27 cm If the radius of the circle is 16 meters, what is the area of sector? 45 hapter Skills Practice 883

78 . If the radius of the circle is 15 feet, what is the area of sector EF? E 30 F 12. If the radius of the circle is 10 inches, what is the area of sector GH? 20 G H 884 hapter Skills Practice

79 Name ate 13. If the radius of the circle is 32 centimeters, what is the area of sector IJ? I J If the radius of the circle is 20 millimeters, what is the area of sector KL? 80 L K hapter Skills Practice 885

80 15. If the radius of the circle is 24 centimeters, what is the area of sector MN? M 135 N 16. If the radius of the circle is 21 meters, what is the area of sector PQ? P 150 Q 886 hapter Skills Practice

81 Name ate alculate the area of each segment. Round your answer to the nearest tenth, if necessary. Use 3.14 to estimate. 17. If the radius of the circle is 6 centimeters, what is the area of the shaded segment? 90 Total area of the circle (6 2 ) 36 cm 2 Sector s fraction of the circle 90º 1 360º 4 rea of sector 1 (36 ) 9 cm2 4 rea of 1 2 (6 6) 18 cm 2 rea of the segment: The area of the shaded segment is approximately 10.3 cm If the radius of the circle is 14 inches, what is the area of the shaded segment? 90 hapter Skills Practice 887

82 19. If the radius of the circle is 17 feet, what is the area of the shaded segment? E 90 F 20. If the radius of the circle is 22 centimeters, what is the area of the shaded segment? 90 G H 888 hapter Skills Practice

83 Name ate 21. If the radius of the circle is 25 meters, what is the area of the shaded segment? I J If the radius of the circle is 30 centimeters, what is the area of the shaded segment? 90 K L hapter Skills Practice 889

84 In circle below, m 90. Use the given information to determine the length of the radius of circle. o 23. If the area of the segment is square feet, what is the length of the radius of circle? 16 90º 360º ( r2 ) ( r2 ) 64 r 2 8 r The length of the radius is 8 feet. 24. If the area of the segment is square inches, what is the length of the radius of circle? 25. If the area of the segment is 2 square meters, what is the length of the radius of circle? 890 hapter Skills Practice

85 Name ate 26. If the area of the segment is square yards, what is the length of the radius of circle? 27. If the area of the segment is square feet, what is the length of the radius of circle? 28. If the area of the segment is square millimeters, what is the length of the radius of circle? hapter Skills Practice 891

86 892 hapter Skills Practice

87 Skills Practice Skills Practice for Lesson.9 Name ate The oordinate Plane ircles and Polygons on the oordinate Plane Problem Set Use the given information to show that each statement is true. Justify your answers by using theorems and by using algebra. 1. The center of circle is at the origin. The coordinates of the given points are ( 4, 0), (4, 0), and (0, 4). Show that Δ is a right triangle. is an inscribed triangle in circle with the hypotenuse as the diameter of the circle, therefore the triangle is a right triangle by the iameter-right Triangle onverse Theorem. ( 4 4 ) 2 (0 0 ) 2 ( 8) ( 4 0 ) 2 (0 4 ) 2 ( 4) 2 ( 4) (4 0 ) 2 (0 4 ) ( 4) (4 2 ) 2 (4 2 ) y x hapter Skills Practice 893

88 2. The center of circle is at the origin. Lines Z and T are tangent to circle. The coordinates of the given points are ( 10, 30), T(8, 6), and Z( 10, 0). Show that the lengths of T and Z are equal y 10 T Z x The center of circle is at the origin. Line is tangent to circle at (3, 4). Show that the tangent line is perpendicular to. y 8 ( 1, 7) 6 4 (3, 4) x (0, 0) hapter Skills Practice

89 Name ate 4. The center of circle is at the origin. The coordinates of the given points are (3, 4), ( 4, 3), (0, 5), E( 3, 4), and F( 1.5, 0.5). Show that EF F F F. E y x F hapter Skills Practice 895

90 5. The center of circle is at the origin. The coordinates of the given points are ( 5, 5), ( 4, 3), (0, 5), and (0, 5). Show that 2. y x hapter Skills Practice

91 Name ate 6. The center of circle is at the origin. hord is perpendicular to radius at point. The coordinates of the given points are ( 5, 0), ( 4, 3), ( 4, 0), and ( 4, 3). Show that bisects y x hapter Skills Practice 897

92 lassify the polygon formed by connecting the midpoints of the sides of each quadrilateral. Show all your work. 7. The rectangle shown has vertices (0, 0), (x, 0), (x, y), and (0, y). Midpoint : U ( x 2, 0 ) Midpoint : S ( x, y 2 ) Midpoint : T ( x 2, y ) Midpoint : R ( 0, y 2 ) y y y Slope RT 2 2 y x 0 x x Slope 2 2 Slope US 0 y y 2 2 x x x y x 2 2 Slope y TS y RU 2 y x x x 2 (0, y) R T (x, y) (0, 0) U (x, 0) y 2 x 2 y y x 2 x y x 2 RU are parallel since they have the same slope of y x. Sides RT and US are parallel since they have the same slope of y x. Sides TS and There are no perpendicular sides. The slopes are not negative reciprocals. RT ( 0 x 2 ) 2 ( y 2 y ) 2 TS ( x 2 x ) 2 ( y y 2 ) 2 ( x 2 ) 2 ( y 2 ) 2 ( x 2 ) 2 ( y 2 ) 2 x 2 y2 4 4 x 2 y2 4 4 US ( x x 2 ) 2 ( y 2 0 ) 2 RU ( 0 x 2 ) 2 ( y 2 0 ) 2 ( x 2 ) 2 ( y 2 ) 2 ( x 2 ) 2 ( y 2 ) 2 x 2 y2 4 4 x 2 y2 4 4 S ll four sides of the new quadrilateral are congruent. pposite sides are parallel and all sides are congruent, so the quadrilateral formed by connecting the midpoints of the rectangle is a rhombus. 898 hapter Skills Practice

93 Name ate 8. The isosceles trapezoid shown has vertices (0, 0), (6, 0), (4, 3), and (2, 3). (2, 3) (4, 3) (0, 0) (6, 0) hapter Skills Practice 899

94 9. The parallelogram shown has vertices (4, 5), (7, 5), (3, 0), and (0, 0). y x hapter Skills Practice

95 Name ate 10. The rhombus shown has vertices (4, 10), (8, 5), (4, 0), and (0, 5). y x hapter Skills Practice 901

96 . The square shown has vertices (0, 0), (4, 0), (4, 4), and (0, 4). y x hapter Skills Practice

97 Name ate 12. The kite shown has vertices (0, 2), (2, 0), (0, 3), and ( 2, 0). y x hapter Skills Practice 903

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