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5.4
Use Medians and Altitudes
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.
D
B
E
P
A
F
C
The medians of ABC meet at P and
2
2
2
AE BP  _____,
CD
BF and CP  _____.
AP  _____,
3
3
3
5.4
Use Medians and Altitudes
G
Example 1 Use the centroid of a triangle
In FGH, M is the centroid and GM = 6.
Find ML and GL.
J
2
_____
GM = GL
____
3
2
6 = GL
___
____
3
9 = GL
___
F
Concurrency of Medians
a Triangle Theorem
Substitute ___
6 for GM.
6
K
M
L
H
of
3
Multiple each side by the reciprocal, ___.
2
Then ML = GL – ____
6 = ___.
3
GM = ___
9 – ____
So, ML = ___
3 and GL = ___.
9
5.4
Use Medians and Altitudes
Checkpoint. Complete the following exercises.
G
1. In Example 1, suppose FM = 10.
Find MK and FK.
2
3
3
 FM  FK
3
2
2
15  FK
MK  FK  FM
 15 10
5
MK  5
FK  15
K
J
M
10
10
F
L
H
5.4
Use Medians and Altitudes
Example 2 Find the centroid of a triangle
The vertices of JKL are J(1, 2), K(4, 6), and L(7, 4).
Find the coordinates of the centroid P of JKL.
Sketch JKL. Then use the Midpoint Formula to find the
midpoint M of JL and sketch median KM.
 1 7 2  4 
M4, 3
M
,
  _______
2 
 2
The centroid is _________
two thirds of the
distance from each vertex to the
midpoint of the opposite side.
The distance from vertex K to point
M is 6 – ___
3 units.
3 = ___
K
L
P
M
J
2
So, the centroid is ___(___)
3 = ___
2 units down from K on KM.
3
The coordinates of the centroid P are (4, 6 – ___),
4, 4
2 or (____).
5.4
Use Medians and Altitudes
Theorem 5.9: Concurrency of Altitudes of a Triangle
The lines containing the altitudes of
a triangle are ___________.
concurrent
G
E
D
The lines containing AF, BE,
and CD meet at G
C
A
F
B
5.4
Use Medians and Altitudes
Example 3 Find the orthocenter
Find the orthocenter P of the triangle.
a.
b.
Solution
a.
b.
P
P
5.4
Use Medians and Altitudes
Checkpoint. Complete the following exercises.
2. In Example 2, where do you need to move point K
so that the centroid is P(4, 5)?
Distance from the midpoint to the
centroid is how much of the total
distance of the median? one third
K
If that distance is 2, what is the
total distance?
P
6
K 4, 9 
M
J
L
5.4
Use Medians and Altitudes
Checkpoint. Complete the following exercises.
3. Find the orthocenter P of the triangle.
P
5.4
Use Medians and Altitudes
Pg. 294, 5.4 #1-19