Section 5.4 Use Medians and Altitudes
Goal  Use medians and altitudes of triangles.
TERM
MEANING
PICTURE
Median
Segment from a vertex to the
midpoint of the opposite side.
BD is a median of ∆ABC
Centroid
Concurrency of the medians of a
triangle
Theorem 5.8 Concurrency of Medians of a Triangle
The medians of a triangle intersect at a point that is two thirds of the
distance from each vertex to the midpoint of the opposite side.
The medians of ∆ABC meet at P and
AP = ⅔AE
Example 1: Use the centroid in a triangle
a. In ΔFGH, M is the centroid and GM = 6.
Find ML and GL.
BP = ⅔BF
CP = ⅔CD
b. In ABC, D is the centroid and BD = 12.
Find DG and BG.
Example 2: In PQR, S is the centroid, PQ  RQ , UQ = 5, TR= 3, and SU = 2.
a. Find RU and RS.
b. Find the perimeter of PQR.
Section 5.4 Use Medians and Altitudes
Example 3: Find the coordinates of the centroid of the triangle with the given vertices.
Step 1: Use the Midpoint Formula.
Step 2: The centroid is two thirds of the distance from
each vertex to the midpoint of the opposite sides. So find
the distance from the vertex to the midpoint.
TERM
MEANING
PICTURE
A perpendicular segment
from a vertex to the opposite
side (or a line that contains
the opposite side)
Altitude
Orthocenter
Concurrency of the altitudes
of a triangle
Theorem 5.9 Concurrency of Altitudes of a Triangle
The lines containing the altitudes of a triangle are concurrent.
Checkpoint: G is the centroid of ∆ABC, AD = 15, CG = 13, AB = AC = 19.2, CE = BF, and AD is an
altitude of ∆ABC. A
F
a. AF
b. FC
c. AE
d. EB
e. AG
f. GD
g. CE
h. GE
i. CD
j. DB
k. GB
l. BF
E
G
C
D
B