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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

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 ___.

Checkpoint. Complete the following exercises.

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

10

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 (____).

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

Find the orthocenter

Example 3

Find the orthocenter P of the triangle.

Solution

P

P

Checkpoint. Complete the following exercises.

- 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

Checkpoint. Complete the following exercises.

- Find the orthocenter P of the triangle.

P

Pg. 294, 5.4 1-19

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