5.1.2 Medians, Altitudes, and Midsegments in Triangles Key Objectives • Apply properties of medians, altitudes, and midsegments of a triangle. Key Terms • A median of a triangle is a segment whose endpoints are a vertex of the triangle and the midpoint of the other side. • The point of concurrency of the medians of a triangle is the centroid of the triangle. • An altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side. • The point of concurrency of the altitudes of a triangle is the orthocenter of a triangle . • A midsegment of a triangle is a segment that joins the midpoints of the two sides of the triangle. • Every triangle has three midsegments, which form the midsegment triangle . Theorems, Postulates, Corollaries, and Properties • Centroid Theorem A triangle’s centroid is located 2/3 of the distance from each vertex to the midpoint of the opposite side. • Triangle Midsegment Theorem A midsegment of a triangle is parallel to a side of the triangle, and its length is half the length of that side. Medians, altitudes, and midsegments are types of line segments in triangles. Medians and altitudes are similar in that each has an endpoint at a vertex and the other endpoint is on the opposite side of the triangle (or on the line containing the opposite side). A midsegment does not have an endpoint at a vertex. Instead, both of a midsegment’s endpoints are on sides of the triangle. A midsegment extends from the midpoint of one side of the triangle to the midpoint of another side of the triangle. Each triangle has three medians, three altitudes, and three midsegments. A triangle’s three medians intersect at one point, the centroid. Similarly, a triangle’s three altitudes also intersect at one point, the orthocenter. However, a triangle’s three midsegments do not intersect at one point. Each midsegment does share an endpoint with another midsegment and the three midsegments form another triangle, the midsegment triangle. Example 1 Using Triangle Medians
PC is given to be a median. Therefore, by definition of median, P is the midpoint of AB . It follows that AP ≅ PB , and so AP = PB . Expressions for AP and PB are given. Set these expressions equal to each other and solve for x . Then, substitute the value of x into the expression for PB to find PB .
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