Unique Triangle!

We are looking for the "least" Pythagorean triangle such that the altitude from the hypotenuse is integer-valued. Letting the sides of the triangle be $a,b,$ the hypotenuse $c$ and the altitude from the hypotenuse be $d,$ then the area of the triangle is both $\frac{1}{2}ab$ and $\frac{1}{2}cd.$ Thus we require that $ab = cd \Longrightarrow d = \dfrac{ab}{c}.$

Now the lengths of a primitive Pythagorean triangle are coprime, which means that for such triangles $d$ cannot be an integer. But if we look at non-primitive Pythagorean triangles of side lengths $ka,kb,kc$ for some integer $k \gt 1$ then $d = \dfrac{kab}{c},$ which implies that $d$ will be an integer if $c|k.$

Now the "least" hypotenuse of any Pythagorean triangle is $5,$ so we will obtain the desired triangle by choosing $k = 5$ and the primitive Pythagorean triangle with side lengths $3,4,5.$ This gives us $15,20,25$ as the side lengths of the desired triangle, rendering a perimeter of $\boxed{60}.$