Construct perpendicular and angle bisectors Use bisectors to solve problems Identify the circumcenter and incenter of a triangle

1. Obj. 20 Bisectors
The student is able to (I can):
• Construct perpendicular and angle bisectors
• Use bisectors to solve problems
• Identify the circumcenter and incenter of a triangle

2. Thm 5-1-1 Perpendicular Bisector Theorem
If a point is on the perpendicular
bisector of a segment, then it is
equidistant from the endpoints of the
segment.
P
D
A
E
PD = AD
PE = AE

3. Thm 5-1-2 Converse of Perp. Bisector Theorem
If a point is equidistant from the
endpoints of a segment, then it is on the
perpendicular bisector of the segment.
S
K
T Y
ST = YT

4. Examples Find each measure:
1. YO
YO = BO = 15
2. GR
B
O
Y
15
G
I
R
20 20
2x-1 x+8
L
2x — 1 = x + 8
x = 9
GR = 2x — 1 + x + 8 = 34

5. Thm 5-1-3
Thm 5-1-4
Angle Bisector Theorem
If a point is on the bisector of an angle,
then it is equidistant from the sides of
the angle.
A
N AN = GN
L G
Converse of the Angle Bisector Theorem
If a point is equidistant from the sides
of an angle, then it is on the angle
bisector.
ÐALN @ ÐGLN

6. circumcenter The intersection of the perpendicular
bisectors of a triangle.

7. circumcenter The intersection of the perpendicular
bisectors of a triangle.
It is called the circumcenter, because it is
the center of a circle that cccciiiirrrrccccuuuummmmssssccccrrrriiiibbbbeeeessss
the triangle (all three vertices are on the
circle).

8. incenter The intersection of the angle bisectors of a
triangle.

9. incenter The intersection of the angle bisectors of a
triangle.
It is called the incenter because it is the
center of the circle that is iiiinnnnssssccccrrrriiiibbbbeeeedddd in the
circle (the circle just touches all three
sides).