A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. You are given a triangle with side lengths of 11 cm, 13 cm, and 16 cm. Determine the length of the median that joins the side of length 16 cm and the opposite vertex.

Source: mathcontest.olemiss.edu 11/2/2009

**SOLUTION**

Let . Let be the median joining vertex to the midpoint . From vertex draw altitude to side .

**Length of **

In right triangle , by the Pythagorean theorem we have

In right triangle , we have by the same theorem

Multiplying the second equation by then adding the two equations yields
—————————–

Since , rewrite the above equation as follows:

**Calculate **

In right triangle :
Thus,

**Length of median **
In right triangle :

Thus, .

**Answer**: 9.

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## About mvtrinh

Retired high school math teacher.

You might enjoy a very different solution based on what I call an “almost pythagorean” relationship that exists in all triangles.

Thank you very much for the alternative solution. Your suggestion has certainly enlarged my knowledge of geometry. The almost pythagorean relationship is quite unexpected and interesting. I have not encountered it in any book during my highschool years. Thanks again.