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# Geometry Section 5-1 1112

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Bisectors of Triangles

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### Geometry Section 5-1 1112

1. 1. Chapter 5 Relationships in TrianglesTuesday, February 28, 2012
2. 2. SECTION 5-1 Bisectors of TrianglesTuesday, February 28, 2012
3. 3. Essential Questions How do you identify and use perpendicular bisectors in triangles? How do you identify and use angle bisectors in triangles?Tuesday, February 28, 2012
4. 4. Vocabulary 1. Perpendicular Bisector: 2. Concurrent Lines: 3. Point of Concurrency: 4. Circumcenter: 5. Incenter:Tuesday, February 28, 2012
5. 5. Vocabulary 1. Perpendicular Bisector: A segment that not only cuts another segment in half, but it also forms a 90° angle at the intersection 2. Concurrent Lines: 3. Point of Concurrency: 4. Circumcenter: 5. Incenter:Tuesday, February 28, 2012
6. 6. Vocabulary 1. Perpendicular Bisector: A segment that not only cuts another segment in half, but it also forms a 90° angle at the intersection 2. Concurrent Lines: Three or more lines that intersect at the same point 3. Point of Concurrency: 4. Circumcenter: 5. Incenter:Tuesday, February 28, 2012
7. 7. Vocabulary 1. Perpendicular Bisector: A segment that not only cuts another segment in half, but it also forms a 90° angle at the intersection 2. Concurrent Lines: Three or more lines that intersect at the same point 3. Point of Concurrency: The common point where three or more lines intersect 4. Circumcenter: 5. Incenter:Tuesday, February 28, 2012
8. 8. Vocabulary 1. Perpendicular Bisector: A segment that not only cuts another segment in half, but it also forms a 90° angle at the intersection 2. Concurrent Lines: Three or more lines that intersect at the same point 3. Point of Concurrency: The common point where three or more lines intersect 4. Circumcenter: The concurrent point where the perpendicular bisectors of the sides of a triangle meet 5. Incenter:Tuesday, February 28, 2012
9. 9. Vocabulary 1. Perpendicular Bisector: A segment that not only cuts another segment in half, but it also forms a 90° angle at the intersection 2. Concurrent Lines: Three or more lines that intersect at the same point 3. Point of Concurrency: The common point where three or more lines intersect 4. Circumcenter: The concurrent point where the perpendicular bisectors of the sides of a triangle meet 5. Incenter: The concurrent point where the angle bisectors of the angles of a triangle meetTuesday, February 28, 2012
10. 10. 5.1 - Perpendicular Bisector Theorem If a point lies on the perpendicular bisector of a segment, the it is equidistant from the endpoints of the segmentTuesday, February 28, 2012
11. 11. 5.1 - Perpendicular Bisector Theorem If a point lies on the perpendicular bisector of a segment, the it is equidistant from the endpoints of the segmentTuesday, February 28, 2012
12. 12. 5.1 - Perpendicular Bisector Theorem If a point lies on the perpendicular bisector of a segment, the it is equidistant from the endpoints of the segment AC = BCTuesday, February 28, 2012
13. 13. 5.2 - Converse of the Perpendicular Bisector Theorem If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segmentTuesday, February 28, 2012
14. 14. 5.2 - Converse of the Perpendicular Bisector Theorem If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segmentTuesday, February 28, 2012
15. 15. 5.2 - Converse of the Perpendicular Bisector Theorem If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment If WX = WZ, then XY = ZYTuesday, February 28, 2012
16. 16. 5.3 - Circumcenter Theorem The circumcenter (concurrent point where perpendicular bisectors intersect) is equidistant from the vertices of a triangleTuesday, February 28, 2012
17. 17. 5.3 - Circumcenter Theorem The circumcenter (concurrent point where perpendicular bisectors intersect) is equidistant from the vertices of a triangleTuesday, February 28, 2012
18. 18. 5.3 - Circumcenter Theorem The circumcenter (concurrent point where perpendicular bisectors intersect) is equidistant from the vertices of a triangle If G is the circumcenter, then GA = GB = GCTuesday, February 28, 2012
19. 19. 5.4 - Angle Bisector Theorem If a point is on the bisector of an angle, then it is equidistant from the sides of the angleTuesday, February 28, 2012
20. 20. 5.4 - Angle Bisector Theorem If a point is on the bisector of an angle, then it is equidistant from the sides of the angleTuesday, February 28, 2012
21. 21. 5.4 - Angle Bisector Theorem If a point is on the bisector of an angle, then it is equidistant from the sides of the angle If AD bisects ∠BAC, BD ⊥ AB, and CD ⊥ AC, then BD = CDTuesday, February 28, 2012
22. 22. 5.5 - Converse of the Angle Bisector Theorem If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angleTuesday, February 28, 2012
23. 23. 5.5 - Converse of the Angle Bisector Theorem If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angleTuesday, February 28, 2012
24. 24. 5.5 - Converse of the Angle Bisector Theorem If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle If BD ⊥ AB, CD ⊥ AC, and BD = CD, then AD bisects ∠BACTuesday, February 28, 2012
25. 25. 5.6 - Incenter Theorem The incenter (concurrent point where angle bisectors meet) is equidistant from each side of the triangleTuesday, February 28, 2012
26. 26. 5.6 - Incenter Theorem The incenter (concurrent point where angle bisectors meet) is equidistant from each side of the triangleTuesday, February 28, 2012
27. 27. 5.6 - Incenter Theorem The incenter (concurrent point where angle bisectors meet) is equidistant from each side of the triangle If S is the incenter of ∆MNP, then RS = TS = USTuesday, February 28, 2012
28. 28. Example 1 Find each measure. a. BC b. XYTuesday, February 28, 2012
29. 29. Example 1 Find each measure. a. BC b. XY BC = 8.5Tuesday, February 28, 2012
30. 30. Example 1 Find each measure. a. BC b. XY BC = 8.5 XY = 6Tuesday, February 28, 2012
31. 31. Example 1 Find each measure. c. PQTuesday, February 28, 2012
32. 32. Example 1 Find each measure. c. PQ 3x + 1 = 5x − 3Tuesday, February 28, 2012
33. 33. Example 1 Find each measure. c. PQ 3x + 1 = 5x − 3 -3x -3xTuesday, February 28, 2012
34. 34. Example 1 Find each measure. c. PQ 3x + 1 = 5x − 3 -3x +3 -3x +3Tuesday, February 28, 2012
35. 35. Example 1 Find each measure. c. PQ 3x + 1 = 5x − 3 -3x +3 -3x +3 4 = 2xTuesday, February 28, 2012
36. 36. Example 1 Find each measure. c. PQ 3x + 1 = 5x − 3 -3x +3 -3x +3 4 = 2x x=2Tuesday, February 28, 2012
37. 37. Example 1 Find each measure. c. PQ 3x + 1 = 5x − 3 PQ = 3x + 1 -3x +3 -3x +3 4 = 2x x=2Tuesday, February 28, 2012
38. 38. Example 1 Find each measure. c. PQ 3x + 1 = 5x − 3 PQ = 3x + 1 -3x +3 -3x +3 PQ = 3(2) + 1 4 = 2x x=2Tuesday, February 28, 2012
39. 39. Example 1 Find each measure. c. PQ 3x + 1 = 5x − 3 PQ = 3x + 1 -3x +3 -3x +3 PQ = 3(2) + 1 4 = 2x PQ = 6 + 1 x=2Tuesday, February 28, 2012
40. 40. Example 1 Find each measure. c. PQ 3x + 1 = 5x − 3 PQ = 3x + 1 -3x +3 -3x +3 PQ = 3(2) + 1 4 = 2x PQ = 6 + 1 x=2 PQ = 7Tuesday, February 28, 2012
41. 41. Example 2 A triangular shaped garden is shown. Can a fountain be placed at the circumcenter and still be in the garden?Tuesday, February 28, 2012
42. 42. Example 2 A triangular shaped garden is shown. Can a fountain be placed at the circumcenter and still be in the garden?Tuesday, February 28, 2012
43. 43. Example 2 A triangular shaped garden is shown. Can a fountain be placed at the circumcenter and still be in the garden? No, it cannotTuesday, February 28, 2012
44. 44. Question If you have an obtuse triangle, where will the circumcenter be? If you have an acute triangle, where will the circumcenter be? If you have an right triangle, where will the circumcenter be?Tuesday, February 28, 2012
45. 45. Question If you have an obtuse triangle, where will the circumcenter be? It will be outside the triangle If you have an acute triangle, where will the circumcenter be? If you have an right triangle, where will the circumcenter be?Tuesday, February 28, 2012
46. 46. Question If you have an obtuse triangle, where will the circumcenter be? It will be outside the triangle If you have an acute triangle, where will the circumcenter be? It will be inside the triangle If you have an right triangle, where will the circumcenter be?Tuesday, February 28, 2012
47. 47. Question If you have an obtuse triangle, where will the circumcenter be? It will be outside the triangle If you have an acute triangle, where will the circumcenter be? It will be inside the triangle If you have an right triangle, where will the circumcenter be? It will be on the hypotenuse of the triangleTuesday, February 28, 2012
48. 48. Example 3 Find each measure. a. DB b. m∠WYZ m∠WYX = 28°Tuesday, February 28, 2012
49. 49. Example 3 Find each measure. a. DB b. m∠WYZ m∠WYX = 28° DB = 5Tuesday, February 28, 2012
50. 50. Example 3 Find each measure. a. DB b. m∠WYZ m∠WYX = 28° DB = 5 m∠WYZ = 28°Tuesday, February 28, 2012
51. 51. Example 3 Find each measure. c. QSTuesday, February 28, 2012
52. 52. Example 3 Find each measure. c. QS 4x - 1 = 3x + 2Tuesday, February 28, 2012
53. 53. Example 3 Find each measure. c. QS 4x - 1 = 3x + 2 -3x -3xTuesday, February 28, 2012
54. 54. Example 3 Find each measure. c. QS 4x - 1 = 3x + 2 -3x +1 -3x +1Tuesday, February 28, 2012
55. 55. Example 3 Find each measure. c. QS 4x - 1 = 3x + 2 -3x +1 -3x +1 x=3Tuesday, February 28, 2012
56. 56. Example 3 Find each measure. c. QS 4x - 1 = 3x + 2 QS = 4x - 1 -3x +1 -3x +1 x=3Tuesday, February 28, 2012
57. 57. Example 3 Find each measure. c. QS 4x - 1 = 3x + 2 QS = 4x - 1 -3x +1 -3x +1 QS = 4(3) - 1 x=3Tuesday, February 28, 2012
58. 58. Example 3 Find each measure. c. QS 4x - 1 = 3x + 2 QS = 4x - 1 -3x +1 -3x +1 QS = 4(3) - 1 x=3 QS = 12 - 1Tuesday, February 28, 2012
59. 59. Example 3 Find each measure. c. QS 4x - 1 = 3x + 2 QS = 4x - 1 -3x +1 -3x +1 QS = 4(3) - 1 x=3 QS = 12 - 1 QS = 11Tuesday, February 28, 2012
60. 60. Example 4 Find each measure if S is the incenter of ∆MNP. a. SUTuesday, February 28, 2012
61. 61. Example 4 Find each measure if S is the incenter of ∆MNP. a. SU SU is a leg in a right triangleTuesday, February 28, 2012
62. 62. Example 4 Find each measure if S is the incenter of ∆MNP. a. SU SU is a leg in a right triangle a +b =c 2 2 2Tuesday, February 28, 2012
63. 63. Example 4 Find each measure if S is the incenter of ∆MNP. a. SU SU is a leg in a right triangle a +b =c 2 2 2 a + 8 = 10 2 2 2Tuesday, February 28, 2012
64. 64. Example 4 Find each measure if S is the incenter of ∆MNP. a. SU SU is a leg in a right triangle a +b =c 2 2 2 a + 8 = 10 2 2 2 a + 64 = 100 2Tuesday, February 28, 2012
65. 65. Example 4 Find each measure if S is the incenter of ∆MNP. a. SU SU is a leg in a right triangle a +b =c 2 2 2 a + 8 = 10 2 2 2 a + 64 = 100 2 a = 36 2Tuesday, February 28, 2012
66. 66. Example 4 Find each measure if S is the incenter of ∆MNP. a. SU SU is a leg in a right triangle a +b =c 2 2 2 a + 8 = 10 2 2 2 a + 64 = 100 2 a = 36 2 a=6Tuesday, February 28, 2012
67. 67. Example 4 Find each measure if S is the incenter of ∆MNP. a. SU SU is a leg in a right triangle a +b =c 2 2 2 a + 8 = 10 2 2 2 a + 64 = 100 2 a = 36 2 a=6 SU = 6Tuesday, February 28, 2012
68. 68. Example 4 Find each measure if S is the incenter of ∆MNP. b. m∠SPUTuesday, February 28, 2012
69. 69. Example 4 Find each measure if S is the incenter of ∆MNP. b. m∠SPU An incenter is created at the concurrent point of the angle bisectorsTuesday, February 28, 2012
70. 70. Example 4 Find each measure if S is the incenter of ∆MNP. b. m∠SPU An incenter is created at the concurrent point of the angle bisectors m∠MNP = 28 + 28 = 56°Tuesday, February 28, 2012
71. 71. Example 4 Find each measure if S is the incenter of ∆MNP. b. m∠SPU An incenter is created at the concurrent point of the angle bisectors m∠MNP = 28 + 28 = 56° m∠NMP = 31 + 31 = 62°Tuesday, February 28, 2012
72. 72. Example 4 Find each measure if S is the incenter of ∆MNP. b. m∠SPU An incenter is created at the concurrent point of the angle bisectors m∠MNP = 28 + 28 = 56° m∠NMP = 31 + 31 = 62° m∠MPN = 180 − 62 − 56 = 62°Tuesday, February 28, 2012
73. 73. Example 4 Find each measure if S is the incenter of ∆MNP. b. m∠SPU An incenter is created at the concurrent point of the angle bisectors m∠MNP = 28 + 28 = 56° m∠NMP = 31 + 31 = 62° m∠MPN = 180 − 62 − 56 = 62° 1 m∠SPU = (62) = 31° 2Tuesday, February 28, 2012
74. 74. Example 4 Find each measure if S is the incenter of ∆MNP. b. m∠SPU An incenter is created at the concurrent point of the angle bisectors m∠MNP = 28 + 28 = 56° m∠NMP = 31 + 31 = 62° m∠MPN = 180 − 62 − 56 = 62° 1 m∠SPU = (62) = 31° 2 Check: 28 + 28 + 31 + 31 + 31 + 31 = 180Tuesday, February 28, 2012
75. 75. Check Your Understading Make sure to review p. 327 #1-8Tuesday, February 28, 2012
76. 76. Problem SetTuesday, February 28, 2012
77. 77. Problem Set p. 327 #9-29 odd, 48 "Great opportunities to help others seldom come, but small ones surround us every day." - Sally KochTuesday, February 28, 2012