1. Chapter 5
Relationships in Triangles
Tuesday, February 28, 2012
2. SECTION 5-1
Bisectors of Triangles
Tuesday, February 28, 2012
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. Vocabulary
1. Perpendicular Bisector:
2. Concurrent Lines:
3. Point of Concurrency:
4. Circumcenter:
5. Incenter:
Tuesday, February 28, 2012
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. 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. 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. 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. 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 meet
Tuesday, February 28, 2012
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 segment
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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 segment
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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 = BC
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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 segment
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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 segment
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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 = ZY
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16. 5.3 - Circumcenter
Theorem
The circumcenter (concurrent point where perpendicular
bisectors intersect) is equidistant from the vertices of a
triangle
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17. 5.3 - Circumcenter
Theorem
The circumcenter (concurrent point where perpendicular
bisectors intersect) is equidistant from the vertices of a
triangle
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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 = GC
Tuesday, February 28, 2012
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 angle
Tuesday, February 28, 2012
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 angle
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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 = CD
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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 angle
Tuesday, February 28, 2012
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 angle
Tuesday, February 28, 2012
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 ∠BAC
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25. 5.6 - Incenter Theorem
The incenter (concurrent point where angle bisectors
meet) is equidistant from each side of the triangle
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26. 5.6 - Incenter Theorem
The incenter (concurrent point where angle bisectors
meet) is equidistant from each side of the triangle
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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 = US
Tuesday, February 28, 2012
28. Example 1
Find each measure.
a. BC b. XY
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29. Example 1
Find each measure.
a. BC b. XY
BC = 8.5
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30. Example 1
Find each measure.
a. BC b. XY
BC = 8.5 XY = 6
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31. Example 1
Find each measure.
c. PQ
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32. Example 1
Find each measure.
c. PQ
3x + 1 = 5x − 3
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33. Example 1
Find each measure.
c. PQ
3x + 1 = 5x − 3
-3x -3x
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34. Example 1
Find each measure.
c. PQ
3x + 1 = 5x − 3
-3x +3 -3x +3
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35. Example 1
Find each measure.
c. PQ
3x + 1 = 5x − 3
-3x +3 -3x +3
4 = 2x
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36. Example 1
Find each measure.
c. PQ
3x + 1 = 5x − 3
-3x +3 -3x +3
4 = 2x
x=2
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37. Example 1
Find each measure.
c. PQ
3x + 1 = 5x − 3 PQ = 3x + 1
-3x +3 -3x +3
4 = 2x
x=2
Tuesday, February 28, 2012
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=2
Tuesday, February 28, 2012
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. 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. Example 2
A triangular shaped garden is shown. Can a fountain be
placed at the circumcenter and still be in the garden?
No, it cannot
Tuesday, February 28, 2012
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. 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. 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. 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 triangle
Tuesday, February 28, 2012
48. Example 3
Find each measure.
a. DB b. m∠WYZ
m∠WYX = 28°
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49. Example 3
Find each measure.
a. DB b. m∠WYZ
m∠WYX = 28°
DB = 5
Tuesday, February 28, 2012
50. Example 3
Find each measure.
a. DB b. m∠WYZ
m∠WYX = 28°
DB = 5 m∠WYZ = 28°
Tuesday, February 28, 2012
51. Example 3
Find each measure.
c. QS
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52. Example 3
Find each measure.
c. QS
4x - 1 = 3x + 2
Tuesday, February 28, 2012
53. Example 3
Find each measure.
c. QS
4x - 1 = 3x + 2
-3x -3x
Tuesday, February 28, 2012
54. Example 3
Find each measure.
c. QS
4x - 1 = 3x + 2
-3x +1 -3x +1
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55. Example 3
Find each measure.
c. QS
4x - 1 = 3x + 2
-3x +1 -3x +1
x=3
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56. Example 3
Find each measure.
c. QS
4x - 1 = 3x + 2 QS = 4x - 1
-3x +1 -3x +1
x=3
Tuesday, February 28, 2012
57. Example 3
Find each measure.
c. QS
4x - 1 = 3x + 2 QS = 4x - 1
-3x +1 -3x +1
QS = 4(3) - 1
x=3
Tuesday, February 28, 2012
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 - 1
Tuesday, February 28, 2012
60. Example 4
Find each measure if S is the incenter of ∆MNP.
a. SU
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61. Example 4
Find each measure if S is the incenter of ∆MNP.
a. SU
SU is a leg in a right triangle
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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 2
Tuesday, February 28, 2012
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 2
Tuesday, February 28, 2012
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
2
Tuesday, February 28, 2012
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
2
Tuesday, February 28, 2012
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=6
Tuesday, February 28, 2012
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 = 6
Tuesday, February 28, 2012
68. Example 4
Find each measure if S is the incenter of ∆MNP.
b. m∠SPU
Tuesday, February 28, 2012
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 bisectors
Tuesday, February 28, 2012
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. 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. 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. 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
Tuesday, February 28, 2012
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 = 180
Tuesday, February 28, 2012
77. Problem Set
p. 327 #9-29 odd, 48
"Great opportunities to help others seldom come, but
small ones surround us every day." - Sally Koch
Tuesday, February 28, 2012