Chapter 5 Lesson 3 Objective: To identify properties of medians and altitudes of a triangle.

A median of a triangle is a segment whose endpoints are a vertex and the midpoint of the opposite side.

Theorem 5-8 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.

In a triangle, the point of concurrency of the medians is the centroid. The point is also called the center of gravity of a triangle because it is the point where a triangular shape will balance.

Example 1: Finding Lengths of Medians D is the centroid of ∆ABC and DE = 6. Find BE. Since D is a centroid, BD =  BE and DE = BE.                                                                                                     

Example 2: Finding Lengths of Medians Use Example 1 to find BD. BD =  BE

An altitude of a triangle is the perpendicular segment from a vertex to the line containing the opposite side. Unlike angle bisectors and medians, an altitude of a triangle can be a side of a triangle or it may lie outside the triangle.

Example 3: Identifying Medians and Altitudes Is ST a median, an altitude, or neither? Explain. ST is an altitude; perpendicular segment from a vertex

Example 4: Identifying Medians and Altitudes Is UW a median, an altitude, or neither? Explain. Median; UW is a segment drawn from vertex U to the midpoint of the opposite side.

The lines that contain the altitudes of a triangle are concurrent. The lines containing the altitudes of a triangle are concurrent at the orthocenter of a triangle. Theorem 5-9   The lines that contain the altitudes of a triangle are concurrent.

Assignment: Page 260 #11-16; 19-22