2.11 Review power point presentation

power point

1. Homework 2.6 Work on Written Math (6 – 10) on page 269.

2. Review on the Concepts of Circle Subtitle

3. E F D O C A B G H J I Circle - set of all points that are of the same distance from a given point in a plane

4. E F D O C A B G H J I The given point is called the center of the circle.

5. E F D O C A B G H J I The segment from the center to any point on the circle is called the radius .

6. E F D O C A B G H J I A segment whose endpoints both lie on the circle is called the chord .

7. E F D O C A B G H J I A chord that passes through the center of the circle is called diameter .

8. E F D O C A B G H J I Interior of the Circle

9. E F D O C A B G H J I Exterior of the Circle

10. E F D O C A B G H J I A secant is a line that intersects the circle in two points.

11. E F D O C A B G H J I A tangent is a line that intersects the circle in exactly one point.

12. O D A B If a radius is perpendicular to a chord, then it bisects the chord. Theorem 97 Given: Circle O with radius at E   E Conclusion: bisects , so AE = BE  

13. P Q M N , MO = 2x + 7, and NO = 3x – 2. Find MO, NO, and MN.   Example 1 O

14. O C A B , AD = x + 6, and CD = x + 8. Find AD, CD, and AC.   Example 2 D

15. O C A B If AG = 12 cm, what is AC? Mental Math G  

16. O C A B If AC = 26 cm, what is CG? Mental Math G  

17. If a radius is perpendicular to a chord, then it bisects the chord. Theorem 97 Theorem 98 If a radius bisects the chord (that is not a diameter), then it is perpendicular to a chord

18. O P A B Q Theorem 99 The perpendicular bisector of a chord passes through the center of the circle.

19. O P A B 5 cm 5 cm OA = PB = 5 cm Congruent Circles Congruent circles are circles that have congruent radii.

20. O C Concentric Circles Concentric circles are coplanar circles having the same center. B A

21. O D A B If chords of a circle (or of congruent circles) are equidistant from the center(s), then the chords are congruent. Theorem 100 E F C Given: Circle O, OC = OF Conclusion:  

22. O D A B If chords of a circle (or of congruent circles) are congruent, then they are equidistant from the center of the circle. Theorem 101 E F C Given: Circle O,   Conclusion: OC = OF

23. B O A C Central Angle - An angle whose vertex is the center of the circle

24. B O A C Minor Arc The minor arc AB is the union of points A and B and all the points of the circle in the interior of the central angle AOB.  

25. B O A C Major Arc The major arc ACB is the union of points A and B and all the points of the circle in the exterior of the central angle AOB.  

26. B O C Semicircle The semicircle is the union of the endpoints of a diameter and all points of the circle that lie on one side of the diameter. D

27. B O A C The degree measure of the minor arc is equal to the measure of the central angle. 40   40   320  

28. B O A C The degree measure of the major arc is equal to 360 minus the degree measure of its related minor arc.   40   40   320  

29. B O C D The degree measure of a semicircle is 180 .   180  

30. M C A The measure of an arc formed by two adjacent, non-overlapping arcs is the sum of the measures of the two arcs. Arc Addition Postulate B D 2 25   40   7   25  

31. F O E G If and , find mEFG .  

32. F O E G If , find mEGF .  

33. M C A - Arcs with the same measure Congruent Arcs B D 40   40  

34. O C A B D Theorem 102 If two minor arcs of a circle or of congruent circles are congruent, then their corresponding chords are congruent.

35. O C A B D Theorem 103 If two chords of a circle or of congruent circles are congruent, then their corresponding minor arcs are congruent.

36. O C A B D Theorem 104 If two central angles of a circle or of congruent circles are congruent, then their corresponding minor arcs are congruent.

37. O C A B D Theorem 105 If two minor arcs of a circle or of congruent circles are congruent, then their corresponding central angles are congruent.

38. O C A B D Theorem 106 If two central angles of a circle or of congruent circles are congruent, then their corresponding chords are congruent.

39. O C A B D Theorem 107 If two chords of a circle or of congruent circles are congruent, then their corresponding central angles are congruent.

40. O C A B In Circle O, . Find the measure of the major arc ABC.   48  

41. Inscribed Angle - An angle whose vertex lies on the circle and whose sides contain chords of a circle

42. Inscribed Angle - An angle whose vertex lies on the circle and whose sides contain chords of a circle

43. Inscribed Angle            

44. An angle inscribed in a semicircle is a right triangle.

45. Corollary 108.2 If two inscribed angles intercept the same arc or congruent arcs, then the angles are congruent.

46. Corollary 108.2 If two inscribed angles intercept the same arc or congruent arcs, then the angles are congruent.  

47. Corollary 108.3 Opposite angles of an inscribed quadrilateral are supplementary.

48. Corollary 108.4 If two arcs of a circle are inscribed between parallel secants, then the arcs are congruent.

53. Theorem 113 If two secants intersect in the interior of a circle, then the measure of the angle formed is equal to one-half the sum of the measures of the intercepted arcs.

54.     a. Find m AEB.   b . Find m AED.  

58. Theorem 114 If two secants intersect in the exterior of a circle, then the measure of the angle formed is equal to one-half the positive difference of the measures of the intercepted arcs.

59. A B C D

60.       A B C D

61.       A B C D

62. Theorem 116 The measure of an angle formed by a secant and a tangent intersecting on the exterior of a circle is equal to one-half the positive difference of the measures of the intercepted arcs.

63. A C B P

64. A C B P      

65. A C B P      

66. Theorem 117 The measure of an angle formed by two tangents to the same circle is equal to one-half the positive difference of the measures of the intercepted arcs.

67. A C B P    

68. T he Power Theorems deal with the products of the lengths of segments related to circles.

69. Theorem 118 If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord.

70.         Theorem 118 If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord.

71.         Theorem 118 If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord.

72.         Theorem 118 If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord.

73. A C B P D Theorem 119 If two secants intersect in the exterior of a circle, then the product of the lengths of one secant segment and its external part equals the product of the lengths of the other secant segments and its external part.

74. A C B P D         Theorem 119 If two secants intersect in the exterior of a circle, then the product of the lengths of one secant segment and its external part equals the product of the lengths of the other secant segments and its external part.

75. A C B P Theorem 120 If a tangent and a secant segment intersect in the exterior of a circle, then the square of the length of the tangent segment is equal to the product of the lengths of the entire secant segment and its external part.

76. A C B P Theorem 120 If a tangent and a secant segment intersect in the exterior of a circle, then the square of the length of the tangent segment is equal to the product of the lengths of the entire secant segment and its external part.      

77. A C B P Theorem 120 If a tangent and a secant segment intersect in the exterior of a circle, then the square of the length of the tangent segment is equal to the product of the lengths of the entire secant segment and its external part.