bisectors in triangles. since a triangle has ________ sides, it has three ___________ ____________...

CHAPTER 5Bisectors in Triangles

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

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

THEOREMS

The circumcenter of ΔABC is the center of its _____________________ circle.

circumscribed

CIRCUMCENTER OF A TRIANGLE

The circumcenter can be inside the triangle, outside the triangle, or on the triangle.

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

EXAMPLE 2MZ is a perpendicular bisector of ∆GHJ.

Find the measures.

ZJ =

ZH =

GM =

KG =

19.9

19.9

14.5

18.6

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

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

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

EXAMPLE 4Find the circumcenter of ∆GOH with vertices G(0, –4), O(0, 0), and H(6, 0) .

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!

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.

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

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