geometry section 5-1 1112
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Chapter 5Relationships in Triangles
Tuesday, February 28, 2012
SECTION 5-1Bisectors of Triangles
Tuesday, February 28, 2012
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
Vocabulary1. Perpendicular Bisector:
2. Concurrent Lines:
3. Point of Concurrency:
4. Circumcenter:
5. Incenter:
Tuesday, February 28, 2012
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:
Tuesday, February 28, 2012
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
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
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
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
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
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
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
Tuesday, February 28, 2012
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
Tuesday, February 28, 2012
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
Tuesday, February 28, 2012
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
Tuesday, February 28, 2012
5.3 - Circumcenter Theorem
The circumcenter (concurrent point where perpendicular bisectors intersect) is equidistant from the vertices of a
triangle
Tuesday, February 28, 2012
5.3 - Circumcenter Theorem
The circumcenter (concurrent point where perpendicular bisectors intersect) is equidistant from the vertices of a
triangle
Tuesday, February 28, 2012
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
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
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
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
Tuesday, February 28, 2012
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
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
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
Tuesday, February 28, 2012
5.6 - Incenter Theorem
The incenter (concurrent point where angle bisectors meet) is equidistant from each side of the triangle
Tuesday, February 28, 2012
5.6 - Incenter Theorem
The incenter (concurrent point where angle bisectors meet) is equidistant from each side of the triangle
Tuesday, February 28, 2012
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
Example 1Find each measure.
a. BC b. XY
Tuesday, February 28, 2012
Example 1Find each measure.
a. BC
BC = 8.5
b. XY
Tuesday, February 28, 2012
Example 1Find each measure.
a. BC
BC = 8.5
b. XY
XY = 6
Tuesday, February 28, 2012
Example 1
c. PQ
Find each measure.
Tuesday, February 28, 2012
Example 1
c. PQ
Find each measure.
3x + 1 = 5x − 3
Tuesday, February 28, 2012
Example 1
c. PQ
Find each measure.
3x + 1 = 5x − 3-3x -3x
Tuesday, February 28, 2012
Example 1
c. PQ
Find each measure.
3x + 1 = 5x − 3-3x -3x +3+3
Tuesday, February 28, 2012
Example 1
c. PQ
Find each measure.
3x + 1 = 5x − 3-3x -3x +3+3
4 = 2x
Tuesday, February 28, 2012
Example 1
c. PQ
Find each measure.
3x + 1 = 5x − 3-3x -3x +3+3
4 = 2xx = 2
Tuesday, February 28, 2012
Example 1
c. PQ
Find each measure.
3x + 1 = 5x − 3-3x -3x +3+3
4 = 2xx = 2
PQ = 3x + 1
Tuesday, February 28, 2012
Example 1
c. PQ
Find each measure.
3x + 1 = 5x − 3-3x -3x +3+3
4 = 2xx = 2
PQ = 3x + 1PQ = 3(2) + 1
Tuesday, February 28, 2012
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
Tuesday, February 28, 2012
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
Tuesday, February 28, 2012
Example 2A triangular shaped garden is shown. Can a fountain be placed at the circumcenter and still be in the garden?
Tuesday, February 28, 2012
Example 2A triangular shaped garden is shown. Can a fountain be placed at the circumcenter and still be in the garden?
Tuesday, February 28, 2012
Example 2A 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
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?
Tuesday, February 28, 2012
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
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
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
Example 3Find each measure.
a. DB b. m∠WYZ
m∠WYX = 28°
Tuesday, February 28, 2012
Example 3Find each measure.
a. DB
DB = 5
b. m∠WYZ
m∠WYX = 28°
Tuesday, February 28, 2012
Example 3Find each measure.
a. DB
DB = 5
b. m∠WYZ
m∠WYZ = 28°
m∠WYX = 28°
Tuesday, February 28, 2012
Example 3
c. QS
Find each measure.
Tuesday, February 28, 2012
Example 3
c. QS
Find each measure.
4x - 1 = 3x + 2
Tuesday, February 28, 2012
Example 3
c. QS
Find each measure.
4x - 1 = 3x + 2-3x -3x
Tuesday, February 28, 2012
Example 3
c. QS
Find each measure.
4x - 1 = 3x + 2-3x -3x +1+1
Tuesday, February 28, 2012
Example 3
c. QS
Find each measure.
4x - 1 = 3x + 2-3x -3x +1+1
x = 3
Tuesday, February 28, 2012
Example 3
c. QS
Find each measure.
4x - 1 = 3x + 2-3x -3x +1+1
x = 3
QS = 4x - 1
Tuesday, February 28, 2012
Example 3
c. QS
Find each measure.
4x - 1 = 3x + 2-3x -3x +1+1
x = 3
QS = 4x - 1QS = 4(3) - 1
Tuesday, February 28, 2012
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
Tuesday, February 28, 2012
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
Tuesday, February 28, 2012
Example 4Find each measure if S is the incenter of ∆MNP.
a. SU
Tuesday, February 28, 2012
Example 4Find each measure if S is the incenter of ∆MNP.
a. SU SU is a leg in a right triangle
Tuesday, February 28, 2012
Example 4Find each measure if S is the incenter of ∆MNP.
a. SU SU is a leg in a right triangle
a2 + b2 = c2
Tuesday, February 28, 2012
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
Tuesday, February 28, 2012
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
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
Tuesday, February 28, 2012
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
Tuesday, February 28, 2012
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
Example 4Find each measure if S is the incenter of ∆MNP.
b. m∠SPU
Tuesday, February 28, 2012
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
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
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
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
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
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
Check Your Understading
Make sure to review p. 327 #1-8
Tuesday, February 28, 2012
Problem Set
Tuesday, February 28, 2012
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
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