Perimeter and area of a right triangle

Question

If I have the perimeter and area of a right triangle, is it possible to find all the three sides of the triangle

Answer 1

Solve the $2\times2$ system

$$b\,h=2A,\\b+h+\sqrt{b^2+h^2}=P.$$

You can work it out as

$$b^4+4A^2=(Pb-b^2-2A)^2.$$

There are two solutions.

Answer 2

If the sides of the triangle are $a,b,c$ and $c$ is the hypotenuse, Solve these 3 equations for the 3 unknown sides:

$\text{Area}=\frac12 ab , \text{Perimeter}=a+b+c \text{ and } a^2+b^2=c^2$

Answer 3

Another try: $$(P-c)^2=(a+b)^2=a^2+b^2+2ab=c^2+4A$$ from where $$c=\frac{P^2-4A}{2P}.$$