4.5 Special Segments in Triangles

* 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
YT
ST= YTKTSY⊥

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 circumscribesthe 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 inscribedin 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 massfor 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
R2
3
GRGY= 2
3
HRHZ= 2
3
JRJX= 2
3

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