bisectors in triangles. since a triangle has ________ sides, it has three ___________ ____________...
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CHAPTER 5Bisectors in Triangles
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PERPENDICULAR BISECTORS IN TRIANGLES
Since a triangle has ________ sides, it has three ___________ ____________
The perpendicular bisector of a side of a _____________ does not always pass through the _____________ ________
3perpendicular
bisectors
triangleopposite
vertex
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VOCABULARY IN CHAPTER 5
When three or more lines intersect at one point, the lines are said to be _________________.
The ___________ ____ ___________ is the point where these lines intersect.
The three perpendicular bisectors of a triangle are ______________. This point of concurrency is the ______________ of the ____________
concurrentpoint of concurrenc
y
concurrent circumcenter
triangle
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THEOREMS
The circumcenter of ΔABC is the center of its _____________________ circle.
circumscribed
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CIRCUMCENTER OF A TRIANGLE
The circumcenter can be inside the triangle, outside the triangle, or on the triangle.
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EXAMPLE 1
DG, EG, and FG are the perpendicular bisectors of ∆ABC. Find GC.
G is the _______________ of ∆ABC. G is equidistant from the vertices of ∆ABC --using the circumcenter theorem
GB =
GA =
GC =
13.4
13.4
13.4 circumcenter
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EXAMPLE 2MZ is a perpendicular bisector of ∆GHJ.
Find the measures.
ZJ =
ZH =
GM =
KG =
19.9
19.9
14.5
18.6
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EXAMPLE 3Find the circumcenter of ∆HJK with vertices H(0, 0), J(10, 0), and K(0, 6).
H
K
J
Step 1---Graph the triangle
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EXAMPLE 3Find the circumcenter of ∆HJK with vertices H(0, 0), J(10, 0), and K(0, 6).
H
K
J
Step 2 – graph the midpoints of each side
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EXAMPLE 3Find the circumcenter of ∆HJK with vertices H(0, 0), J(10, 0), and K(0, 6).
H
K
J
Step 3 – find the slope of each side, then find the perpendicular slope of each side
m HJ =
m HK =
m KJ =
und
zero
5
3
m HK = m HJ =
m KJ =
zero
und
3
5
(5, 3) is the circumcenter of the triangle
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EXAMPLE 4Find the circumcenter of ∆GOH with vertices G(0, –4), O(0, 0), and H(6, 0) .
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INCENTER OF A TRIANGLEA triangle has three angles, so it has three angle bisectors. The angle bisectors of a triangle are also concurrent. This point of concurrency is the incenter of the triangle .
The distance between a point and a line is the length of the perpendicular segment from thepoint to the line.
Remember!
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INCENTERUnlike the circumcenter, the incenter is always inside the triangle.
The incenter is the center of the triangle’s inscribed circle. A circle inscribed in a polygon intersects each line that contains a side of the polygon at exactly one point.
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EXAMPLE 5
MP and LP are angle bisectors of ∆LMN. Find the distance from P to MN.
P is the incenter of ∆LMN. By the Incenter Theorem, P is equidistant from the sides of ∆LMN.
The distance from P to LM is 5.
So the distance from P to MN = ________ .5
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EXAMPLE 5
MP and LP are angle bisectors of ∆LMN. Find mPMN.
mPLN = 50o
mMLN = 100°
mL + mN + mM = 180°
100 + 20 + mM= 180
mM = 60° mPMN= 30o