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What is the "altitude to the hypotenuse," and how do I find its length, if I know the lengths of two legs?

To answer this question, consider the diagram below. I show a triangle with angles A, B, and C, and right angle at C. The segment CD is the altitude to the hypotenuse. An altitude is a line segment that is perpendicular to a side, and passes through the opposite vertex. Thus, the altitude to the hypotenuse is perpendicular to the hypotenuse, and passes through the point C.

For this example, I gave the legs lengths 3 and 4. I did this so it would work out simply: the hypotenuse is 5 (because 32 + 42 = 52).

The question is, how do we find the length of CD? To figure this out, it is helpful to notice that all three triangles are similar. 


From this, we know that the ratio of the longer leg to the hypotenuse is the same for each triangle.

In the largest triangle, that ratio is 4/5.  In the smallest triangle, that ratio is CD/3.

Thus, 4/5 = CD/3, which leads to CD = 12/5.


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