1. 5-4 Medians and Altitudes
Course: Geometry pre-IB
Quarter: 2nd
Objective: Use properties of medians and altitudes of a triangle.
SSS: MA.912.G.4.2, MA.912.G.4.5
Motivating the Lesson: Demonstrate how to balance a cardboard triangle on a pencil. Ask students what they think about the “balancing point” of the triangle (it’s the center). Does every triangle have this balancing point?

2. Medians
A median of a triangle is a segment whose endpoint are a vertex and the midpoint of the opposite side.
The point of concurrency of the medians is the centroid of the triangle (the “balancing” point).

3. Centroid
Concurrency of Medians Theorem: The medians of a triangle are concurrent at a point that is two thirds the distance from each vertex to the midpoint of the opposite side.
DC = ⅔DJ EC = ⅔EG FC = ⅔FH
To find the x-coordinate of the centroid, divide the sum of the x-coordinates of the vertices of the triangle by 3. Do the same with the y-coordinates.

4. Finding a Centroid
The coordinates of ABC are A(7, 10), B(3, 6), and C(5, 2).

5. Altitudes
An altitude of a triangle is the perpendicular segments from a vertex of the triangle to the line containing the opposite side.
Concurrency of Altitudes Theorem: The lines that contain the altitudes of a triangle are concurrent.
The lines that contain the altitudes of a triangle are concurrent at the orthocenter
Can be INSIDE, ON, or OUTSIDE the triangle.

6. Identifying Medians and Altitudes
Is PR a median, altitude, or neither? Explain.
Is QT a median, altitude, or neither? Explain.

7.  For ABC, is each segment a median, altitude, or neither. Explain.AD EG CF

8. “Trick” for Remembering Points of Concurrency
Peanut Butter Cookies
Are Better If
Mom Cooks
After Oprah
Perp. Bisectors—Circumcenter
Angle Bisectors—Incenter
Medians—Centroid
Altitudes—Orthocenter
*Also, the ones “inside” are always “inside” the triangle!