Warm-Up

The perpendicular bisectors meet atG. If BD = 4 and GD = 3, what is the length of GC?

Properties of Triangles – Day 3Medians and Altitudes

Theorem:Concurrency of Medians

The centroid is 2/3 the distance from each vertex to the midpoint of the opposite side.

Example 1:

D is the centroid of the triangle and BE is perpendicular to AC.

Example 2:

Draw a triangle with vertices:

D(3,6), F(7,10), and E(5,2)

Find the midpoint of each side

Find the centroid P

Theorem: Concurrency of Altitudes

The lines containing the altitudes intersect at a point called the orthocenter.

Name the line segment described: Q:Perpendicular segment from vertex to opposite side. A: Altitude Q: Segment that divides an angle of a triangle into two

congruent, adjacent angles. A: Angle Bisector Q: Perpendicular segment that intersects the side of a

triangle at its midpoint. A: Perpendicular Bisector Q: Segment connecting a vertex of a triangle to the

midpoint of the opposite side. A: Median Q: Segment that connects two midpoints of a triangle. A: Midsegment

Name the concurrent points for the following segments:

Q: Angle Bisectors A: Incenter Q: Medians A: Centroid Q: Perpendicular Bisectors A: Circumcenter Q: Altitudes A: Orthocenter

Homework – Day 3

Days 1 – 3 Review

Use your clickers to answer the following questions…

This segment’s endpoints are a vertex of a triangle and the midpoint of the opposite side.

a. Medianb. Perpendicular Bisectorc. Midsegmentd. Altitude

In WXY, Q is the centroid and YQ = 2x 15 and QA = 4. Find x.

a. 9.5b. 11.5c. 13.5

Q

Y

W

X

A

B

C

The circumcenter is equidistant to the _________ of a triangle.

a. Verticesb. Sides

In JKL, PS = 7. Find JP.

a. 7b. 14c. 21 J

K

L

R

S

T

P

This segment is perpendicular to a segment at its midpoint.

a. Medianb. Perpendicular Bisectorc. Midsegmentd. Altitude

This line passes through a vertex and divides that interior angle in half.

a. Perpendicular Bisectorb. Angle Bisectorc. Midsegment

Find the measure of KF if K is the incenter of ABC.

a. 5b. 12c. 13

A

B

C

F

D

E13

12

K

This is the intersection of the three perpendicular bisectors of a triangle and is equidistant from the vertices.a. Incenterb. Circumcenterc. Centroidd. Orthocenter

a. 2b. 4c. 6

D is the centroid of triangle ABC. Find CF.

This is the intersection of the three medians of a triangle and is 2/3 the distance from each vertex to the midpoint of the opposite side.a. Incenterb. Circumcenterc. Centroidd. Orthocenter

This is the intersection of the three angle bisectors of a triangle and is equidistant from the sides.a. Incenterb. Circumcenterc. Centroidd. Orthocenter

The incenter is equidistant to the _________ of a triangle.

a. Verticesb. Sides

This is the intersection of the three altitudes of a triangle.

a. Incenterb. Circumcenterc. Centroidd. Orthocenter

Find each measure of DC if D is the circumcenter of ABC, AD = 12, and DF = 5.

a. 5b. 12c. 13

A

B

C

DE

F

G

This is a perpendicular segment from a vertex to the opposite side.

a. Medianb. Perpendicular Bisectorc. Midsegmentd. Altitude