One of the legs of a right triangle has length 4 cm. Express the length of the altitude perpendicular to the hypotenuse as a function of the length of the hypotenuse

Without loss of generality, we can think of the 4 cm leg of the triangle as being the short leg. It is also the hypotenuse of the smallest of the similar triangles created by the altitude (x). If h represents the hypotenuse of the triangle, then similarity tells us the ratio of the "long" leg to the hypotenuse is the same for the smallest and largest triangles.

... x/4 = √ (h²-4²) / h

Then the altitude (x) as a function of the hypotenuse (h) is ...

... x = (4/h) * √ (h²-16)