How to find the function of the hypotenuse in terms of perimeter?

So we have h = √(a^2 + b^2). Squaring this we get h^2 = a^2 + b^2. Now add 2ab to both sides to get h^2 + 2ab = a^2 + 2ab + b^2. Squaring the right-hand-side we get h^2 + 2ab = (a+b)^2. Substitute the ab on the left-hand-side with ab = 50 to obtain h^2 +100 = (a+b)^2. Let the perimeter be denoted as p = a + b + h, so a + b = p - h. Now substitute this in h^2 + 100 = (a+b)^2 to get h^2 + 100 = (p-h)^2. Expand out the right-hand-side to get h^2 + 100 = p^2 - 2hp + h^2. Cancel h^2 from both sides to get 100 = p^2 - 2hp. Now rearrange to put h in terms of p: (p^2 - 100)/(2p) = (p/2) - (50/p).