Centers of Triangles or Points of

Concurrency

Median

M D P

N

C

In MC and ND are medians.MNP,

What is NC if NP = 18?

9MC bisects NP…so 18/2

If DP = 7.5, find MP.

7.5 + 7.5 = 15

Example 1

How many medians does a triangle have?

The medians of a triangle are concurrent.

The intersection of the medians is

called the CENTRIOD.

Theorem

The length of the segment from the vertex to the

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

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

A

B

M

P E

C

N

If EM = 3, find EC.EC = 2(3)EC = 6

Example 2

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

A

B

M

P E

C

N

If EN = 12, find AN.AE = 2(12)=24

AN = 36

AN = AE + ENAN = 24 + 12

Example 3

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

A

B

M

P E

C

N

If EM = 3x + 4 and CE = 8x, what is x?

x = 4

Example 4

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

A

B

M

P E

C

N

If CM = 24 what is CE?

CE = 2/3CM

CE = 2/3(24)

CE = 16

Example 5

Angle Bisector

X

W

Z Y1

2

Example 1

In WYZ, ZX bisects . If m 1 = 55,

find m .

WZY

WZY

55 55

110

m WZY

m WZY

G

F

H

I

Example 2In FHI, IG is an angle bisector. Find m . HIG

5 1( )x ( )4 1x

5(x – 1) = 4x + 1

5x – 5 = 4x + 1

x = 6

How many angle bisectors does a triangle have?

three

The angle bisectors of a triangle are ____________.concurrent

The intersection of the angle bisectors is called the ________.Incenter

The incenter is the same distance from the sides of the triangle.

B

A C

P

E

D

F

Point P is called the __________.Incenter

Example 4

The angle bisectors of triangle ABC meet at point L. • What segments are congruent?• Find AL and FL.

F

D

E

L

BC

A

8

LF, DL, EL

FL = 6

Triangle ADL is a right triangle, so use

Pythagorean thm

AL2 = 82 + 62

AL2 = 100

AL = 106

Perpendicular Bisector

Example 1: Tell whether each red segment is a perpendicular bisector of the triangle.

Example 2: Find x

3x + 4 5x - 10

How many perpendicular bisectors does a triangle

have?

The perpendicular bisectors of a triangle

are concurrent.

The intersection of the perpendicular bisectors is called

the CIRCUMCENTER.

The Circumcenter is equidistant from the vertices

of the triangle.B

A C

P

PA = PB = PC

6

D

B

A C

P

10

Example 3: The perpendicular bisectors of triangle ABC meet at point P.

• Find DA.• Find BA.• Find PC.• Use the Pythagorean Theorem

to find DP.

DA = 6

BA = 12

PC = 10

DP2 + 62 = 102

DP2 + 36 = 100

DP2 = 64

DP = 8

Altitude

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

The altitude is the “true height” of

the triangle.

How many altitudes does a triangle have?

The altitudes of a triangle are concurrent.

The intersection of the altitudes is called the

ORTHOCENTER.

Tell if the red segment is an altitude, perpendicular bisector, both, or neither?

ALTITUDE

NEITHER

BOTH

PER. BISECTOR