Medians & Altitudes

The intersection of the medians is called the CENTROID.

Theorem 5.8

The length of the segment from the vertex to the

centroid is twice the length of the segment from the centroid to the midpoint.

A BF

X

E

C

D

A BF

X

E

C

D

In ABC, AN, BP, and CM are medians.

A

B

M

P E

C

NIf EM = 3, find EC.

Ex: 1

In ABC, AN, BP, and CM are medians.

A

B

M

P E

C

NIf EN = 12, find AN.

Ex: 2

In ABC, AN, BP, and CM are medians.

A

B

M

P E

C

N

If CM = 3x + 6, and CE = x + 12, what is x?

Ex: 3

Altitude

The intersection of the altitudes is

called the ORTHOCENTER.

Tell whether each red segment is an altitude of the triangle.

The altitude is the “true height” of

the triangle.

Perpendicular Bisector and Angle Bisector

The intersection of the perpendicular bisector is called

the CIRCUMCENTER.

What is special about the

CIRCUMCENTER?

Equidistant to the vertices of the triangle.

Example 2:Point G is the circumcenter of the triangle. Find CG.

B

A

C

G

ED

F

6

8

10

Angle Bisector

The intersection of the angle bisectors is called the INCENTER.

What is special about the INCENTER?

Equidistant to sides of the triangle

Example 1:Point G is the incenter of the triangle. Find GB.

B

A

C

G

ED

F

2

5

7

7

Example 1:

Point N is the incenter of the triangle. Find the length of segment ON.

30 18

Example 2:

Point N is the incenter of the triangle. Find the length of segment NP.

p. 266 #13-18

p. 275 #14-17

p. 280 #1-6, 10-14