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Transcript of Geometry Section 5-1 1112
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Chapter 5Relationships in Triangles
Tuesday, February 28, 2012
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SECTION 5-1Bisectors of Triangles
Tuesday, February 28, 2012
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Essential Questions
How do you identify and use perpendicular bisectors in triangles?
How do you identify and use angle bisectors in triangles?
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Vocabulary1. Perpendicular Bisector:
2. Concurrent Lines:
3. Point of Concurrency:
4. Circumcenter:
5. Incenter:
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Vocabulary1. 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:
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Vocabulary1. 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
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Vocabulary1. 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
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Vocabulary1. 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
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Vocabulary1. 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
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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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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
Tuesday, February 28, 2012
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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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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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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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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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5.3 - Circumcenter Theorem
The circumcenter (concurrent point where perpendicular bisectors intersect) is equidistant from the vertices of a
triangle
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5.3 - Circumcenter Theorem
The circumcenter (concurrent point where perpendicular bisectors intersect) is equidistant from the vertices of a
triangle
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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
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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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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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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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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
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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
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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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5.6 - Incenter Theorem
The incenter (concurrent point where angle bisectors meet) is equidistant from each side of the triangle
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5.6 - Incenter Theorem
The incenter (concurrent point where angle bisectors meet) is equidistant from each side of the triangle
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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
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Example 1Find each measure.
a. BC b. XY
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Example 1Find each measure.
a. BC
BC = 8.5
b. XY
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Example 1Find each measure.
a. BC
BC = 8.5
b. XY
XY = 6
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Example 1
c. PQ
Find each measure.
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Example 1
c. PQ
Find each measure.
3x + 1 = 5x − 3
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Example 1
c. PQ
Find each measure.
3x + 1 = 5x − 3-3x -3x
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Example 1
c. PQ
Find each measure.
3x + 1 = 5x − 3-3x -3x +3+3
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Example 1
c. PQ
Find each measure.
3x + 1 = 5x − 3-3x -3x +3+3
4 = 2x
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Example 1
c. PQ
Find each measure.
3x + 1 = 5x − 3-3x -3x +3+3
4 = 2xx = 2
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Example 1
c. PQ
Find each measure.
3x + 1 = 5x − 3-3x -3x +3+3
4 = 2xx = 2
PQ = 3x + 1
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Example 1
c. PQ
Find each measure.
3x + 1 = 5x − 3-3x -3x +3+3
4 = 2xx = 2
PQ = 3x + 1PQ = 3(2) + 1
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Example 1
c. PQ
Find each measure.
3x + 1 = 5x − 3-3x -3x +3+3
4 = 2xx = 2
PQ = 3x + 1PQ = 3(2) + 1
PQ = 6 + 1
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Example 1
c. PQ
Find each measure.
3x + 1 = 5x − 3-3x -3x +3+3
4 = 2xx = 2
PQ = 3x + 1PQ = 3(2) + 1
PQ = 6 + 1PQ = 7
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Example 2A triangular shaped garden is shown. Can a fountain be placed at the circumcenter and still be in the garden?
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Example 2A triangular shaped garden is shown. Can a fountain be placed at the circumcenter and still be in the garden?
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Example 2A triangular shaped garden is shown. Can a fountain be placed at the circumcenter and still be in the garden?
No, it cannot
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QuestionIf 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?
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QuestionIf 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
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QuestionIf 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
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QuestionIf 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
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Example 3Find each measure.
a. DB b. m∠WYZ
m∠WYX = 28°
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Example 3Find each measure.
a. DB
DB = 5
b. m∠WYZ
m∠WYX = 28°
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Example 3Find each measure.
a. DB
DB = 5
b. m∠WYZ
m∠WYZ = 28°
m∠WYX = 28°
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Example 3
c. QS
Find each measure.
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Example 3
c. QS
Find each measure.
4x - 1 = 3x + 2
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Example 3
c. QS
Find each measure.
4x - 1 = 3x + 2-3x -3x
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Example 3
c. QS
Find each measure.
4x - 1 = 3x + 2-3x -3x +1+1
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Example 3
c. QS
Find each measure.
4x - 1 = 3x + 2-3x -3x +1+1
x = 3
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Example 3
c. QS
Find each measure.
4x - 1 = 3x + 2-3x -3x +1+1
x = 3
QS = 4x - 1
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Example 3
c. QS
Find each measure.
4x - 1 = 3x + 2-3x -3x +1+1
x = 3
QS = 4x - 1QS = 4(3) - 1
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Example 3
c. QS
Find each measure.
4x - 1 = 3x + 2-3x -3x +1+1
x = 3
QS = 4x - 1QS = 4(3) - 1
QS = 12 - 1
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Example 3
c. QS
Find each measure.
4x - 1 = 3x + 2-3x -3x +1+1
x = 3
QS = 4x - 1QS = 4(3) - 1
QS = 12 - 1QS = 11
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Example 4Find each measure if S is the incenter of ∆MNP.
a. SU
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Example 4Find each measure if S is the incenter of ∆MNP.
a. SU SU is a leg in a right triangle
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Example 4Find each measure if S is the incenter of ∆MNP.
a. SU SU is a leg in a right triangle
a2 + b2 = c2
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Example 4Find each measure if S is the incenter of ∆MNP.
a. SU SU is a leg in a right triangle
a2 + b2 = c2
a2 + 82 = 102
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Example 4Find each measure if S is the incenter of ∆MNP.
a. SU SU is a leg in a right triangle
a2 + b2 = c2
a2 + 82 = 102
a2 + 64 = 100
Tuesday, February 28, 2012
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Example 4Find each measure if S is the incenter of ∆MNP.
a. SU SU is a leg in a right triangle
a2 + b2 = c2
a2 + 82 = 102
a2 + 64 = 100
a2 = 36
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Example 4Find each measure if S is the incenter of ∆MNP.
a. SU SU is a leg in a right triangle
a2 + b2 = c2
a2 + 82 = 102
a2 + 64 = 100
a2 = 36
a = 6
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Example 4Find each measure if S is the incenter of ∆MNP.
a. SU SU is a leg in a right triangle
a2 + b2 = c2
a2 + 82 = 102
a2 + 64 = 100
a2 = 36
a = 6SU = 6
Tuesday, February 28, 2012
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Example 4Find each measure if S is the incenter of ∆MNP.
b. m∠SPU
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Example 4Find 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
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Example 4Find 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
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Example 4Find 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
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Example 4Find 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
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Example 4Find 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°
m∠SPU =
12
(62) = 31°
Tuesday, February 28, 2012
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Example 4Find 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°
m∠SPU =
12
(62) = 31°
Check: 28 + 28 + 31 + 31 + 31 + 31 = 180Tuesday, February 28, 2012
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Check Your Understading
Make sure to review p. 327 #1-8
Tuesday, February 28, 2012
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Problem Set
Tuesday, February 28, 2012
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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