section 5-3
Medians & Altitudes of Triangles
median of a triangle - a segment whose endpoints are a vertex of the
triangle and the midpoint of the opposite side.
Every triangle has three medians, and the medians are concurrent.
The point of concurrency of the medians of a triangle is the centroid of the
triangle. The centroid is always inside the triangle. The centroid is also called
the center of gravity because it is the point where a triangular region will balance.
Construction: Centroid of a triangle
B
B
Y
X
B
Y
X
C
A
X
C
Z
Z
Step 1: Construct the midpoints of the
3 sides of the triangle by
using the segment bisector
construction.
problem # 1 - In
Y
P
C A
A
Z
step 3: Label the point where
the 3 medians meet.
It is the centroid.
step 2: Draw segments connecting
the 3 vertices to the opposite
midpoints. These are the
medians.
ABC, AF = 9, and GE = 2.4. Find each length.
B
a. AG
E
F
b. CE
G
A
C
D
problem # 2 - A sculptor is shaping a triangular piece of iron that will balance
on the point of a cone. At what coordinates will the triangular
region balance?
altitude of a triangle - a perpendicular segment from a vertex to the line containing
the opposite side.
Every triangle has three altitudes. An altitude can be inside, outside, or on the triangle.
C
C
A
X
Y
Z
B
X
B
A
C
X
B
A
Z
Y
R
In ΔQRS, altitude QY is inside the triangle,
but RX and SZ are not. Notice that the lines
containing the altitudes are concurrent at P.
This point of concurrency is the orthocenter
of the triangle.
Q
S
problem # 3 - Find the orthocenter of
JKL with vertices J(-4,2), K(-2,6), and L(2,2).
y
10
K(-2,6)
8
6
4
J(-4,2)
L(2,2)
2
0
-10
-8
-6
-4
0
-2
2
4
6
8
10
x
-2
-4
-6
-8
-10
Constructions: Centroid and Orthocenter of a Triangle
Discuss the methods that you would use to construct these two triangle centers.
* See the attachments page for a Geometer's Sketchpad
demonstration of Triangle Centers.