Geometry 5.2 Medians and Altitudes of a Triangle

Median of a triangle – a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side.

Point of Concurrency The three medians of a triangle are concurrent. The point of concurrency of the medians is called the centroid of the triangle. The centroid represents the balancing point of the triangle.

Theorem 5.7: Centroid Thrm 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. If P is the centroid of  ABC, then AP = 2/3AD, BP = 2/3BF, and CP = 2/3CE

Example 1 P is the centroid of QRS. PT = 5. Find RT and RP.

Example 2 C is the centroid of GHJ and HM = 24. Find CM and CH. CH=16

Example 3 Find the centroid of the triangle with coordinates (1,10), (5,0), and (9,5).

Altitude – the perpendicular segment from a vertex to the opposite side or to the line that contains the opposite side.

Point of Concurrency The lines containing the altitudes of a triangle are concurrent and intersect at a point called the orthocenter of the triangle.

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