points of concurrency

Points of Concurrency

MM1G3e Students will be able to find and use

points of concurrency in triangles.

Median of a Triangle

• A segment from one vertex of the triangle to the midpoint of the opposite side.

The intersection of the medians is called the CENTROID.

How many medians does a triangle have?

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.EC = 2(3)

Ex: 1

EC = 6

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

A

B

M

P E

C

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

Ex: 2

AN = 36

AN = AE + ENAN = 24 + 12

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?CM = CE + EM

Ex: 3

x = 8

3x + 6 = (x + 12) + .5(x + 12)3x + 6 = x + 12 + .5x + 63x + 6 = 1.5x + 18

1.5x = 12

Altitude

The intersection of the altitudes is called the ORTHOCENTER.

How many altitudes does a triangle have?

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

The altitude is the “true height” of

the triangle.

In ABC, CE and AD are medians.A E

C

B

DG

1. If CD = 3.25, what is BC?

2. Find AG if DG = 10.

3. If CG = 7, find CE?

Altitude, perpendicular bisector, both, or neither?

6.5

20

10.5

ALTITUDE

NEITHERBOTH

PER. BISECTOR

Homework Answerspage 280 1-6, 10-14

1. 82. 163. 54. 155. 126. 6

10. Yes, yes, yes11. No, no, no12. No, yes, no13. 12, 78o

14. 6.5, 15

The intersection of the perpendicular bisector is called the CIRCUMCENTER.

How many perpendicular

bisectors does a triangle have?

What is special about the

CIRCUMCENTER?

Equidistant to the vertices of the triangle.

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

B

A

C

G

ED

F

2

5

7

GB=7

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

B

A

C

G

ED

F

6

8

CG=10

Angle Bisector

The intersection of the angle bisectors is called the INCENTER.

How many angle bisectors does a triangle have?

What is special about the INCENTER?

Equidistant to sides of the triangle

Example 1:Point N is the incenter of the triangle. Find the length of segment ON.

ON=18

30 18

Example 2:Point N is the incenter of the triangle. Find the length of segment NP.

NP=15

p. 266 #13-18

p. 275 #14-17