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

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