When we draw an altitude from the hypotenuse of a right triangle, we get three pairs of similar triangles that can be used to find missing lengths. An altitude is a segment from one vertex of the triangle to the line containing the opposite side and that is perpendicular to the opposite side. For right triangle  we can draw the altitude . 

Why are triangles , , and  all similar to each other?

Triangles and are similar by the Angle-Angle Triangle Similarity Theorem because angle is in both triangles, and both triangles are right triangles, so angles and are congruent. Triangles and are similar by the Angle-Angle Triangle Similarity Theorem because angle is in both triangles, and both triangles are right triangles, so angles and are congruent. Because triangles and are both similar to triangle , they are also similar to each other.

Because the triangles , , and are all similar, corresponding angles are congruent and pairs of corresponding sides are scaled copies of each other, by the same scale factor. We can use the proportionality of pairs of corresponding side lengths to find missing side lengths. For example, suppose we need to find and know that  and . Because triangle is similar to triangle , we know that . So , and . Or, suppose we need to find and know that  and . Because triangle is similar to triangle , we know that . So , and .