5-4 Medians and Altitudes

Median of a Triangle A segment whose endpoints are a vertex and the midpoint of the opposite side.

A triangle’s three medians are always concurrent, and are always concurrent on the interior of the triangle. In a triangle, the point of concurrency of the medians is the centroid of the triangle. The point is also called the center of gravity of a triangle because it is the point where a triangular shape will balance.

Altitude of a triangle The perpendicular segment from a vertex of the triangle to the line containing the opposite side. An altitude of a triangle can be inside the triangle or outside, or in right triangles it is a side of the triangle.

The lines that contain the altitudes of a triangle are concurrent at the orthocenter of the triangle. The orthocenter of a triangle can be inside, on(for right triangles the orthocenter will be the vertex at the right angle), or outside the triangle.

Orthocenter with coordinate geometry Need to use a system of equations

Circumcenter with coordinate geometry Need to use a system of equations

Centroid with coordinate geometry Average of x’s and y’s