Answer
The solution is
$$h=\frac{4\sqrt{c^2-16}}{c}.$$
Work Step by Step
Let $a=4$ be the one leg of this triangle and $b$ the other leg and $c$ be the hypotenuse. Using the expression for the area of such a triangle written in terms of $a$ and $b$ and in terms of $c$ and $h$ where $h$ is the altitude we have
$$\frac{1}{2}ab=\frac{1}{2}ch\Rightarrow h=\frac{ab}{c}.$$
Now from Pythagorean theorem we have
$$c^2=a^2+b^2\Rightarrow b=\sqrt{c^2-a^2}.$$
Putting this into the expression for altitude
$$h=\frac{a\sqrt{c^2-a^2}}{c}.$$
Putting $a=4$
$$h=\frac{4\sqrt{c^2-16}}{c.}$$
