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

1. Special Segments in Triangles
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
• Identify altitudes and medians of triangles
• Identify the orthocenter and centroid of a triangle
• Use triangle segments to solve problems

2. 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. Converse of Perpendicular 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
Y
T
ST = YT
KT SY
⊥

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

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

6. Angle Bisector Theorem
If a point is on the bisector of an angle, then it is
equidistant from the sides of the angle.
Converse of the Angle Bisector Theorem
If a point is equidistant from the sides of an angle, then it
is on the angle bisector.
A
L G
N AN = GN
ALN  GLN

7. circumcenter – the intersection of the perpendicular
bisectors of a triangle.

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

9. incenter – the intersection of the angle bisectors of a
triangle.

10. 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 inscribed in the circle (the circle just touches all three
sides).

11. median – a segment whose endpoints are a vertex of the
triangle and the midpoint of the opposite side.
altitude – a perpendicular segment from a vertex to the line
containing the opposite side.

12. centroid – the intersection of the medians of a triangle. It is
also the center of mass for the triangle.

13. Centroid Theorem
The centroid of a triangle is located of the distance
from each vertex to the midpoint of the opposite side.
G
H
J
X Y
Z
R
2
3
GR GY
=
2
3
HR HZ
=
2
3
JR JX
=
2
3

14. orthocenter – the intersection of the altitudes of a triangle.