Theorem%205.8:%20Concurrency%20of%20Medians%20of%20a%20Triangle

1
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.
P
2
Use the centroid of a triangle
Example 1
6
_____ ____GL
Concurrency of Medians of a
Triangle Theorem
___ ____GL
Substitute ___ for GM.
___ GL
Multiple each side by the reciprocal, ___.
Then ML GL ____ ___ ____ ___.
So, ML ___ and GL ___.
3
Checkpoint. Complete the following exercises.

1. In Example 1, suppose FM 10. Find MK and FK.

10
4
Find the centroid of a triangle
Example 2
K
L
The centroid is _________ of the distance from
each vertex to the midpoint of the opposite side.
two thirds
P
M
J
The distance from vertex K to point M is 6 ___
___ units.
3
3
So, the centroid is ___(___) ___ units down
from K on KM.
The coordinates of the centroid P are (4, 6
___), or (____).
5
Theorem 5.9 Concurrency of Altitudes of a
Triangle
The lines containing the altitudes of a triangle
are ___________.
G
E
D
The lines containing AF, BE, and CD meet at G
F
6
Find the orthocenter
Example 3
Find the orthocenter P of the triangle.
Solution
P
P
7
Checkpoint. Complete the following exercises.

1. 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?
K
If that distance is 2, what is the total distance?
P
L
M
J
8
Checkpoint. Complete the following exercises.

1. Find the orthocenter P of the triangle.

P
9
Pg. 294, 5.4 1-19