Question

Find the general values of $x$ and $y$ which make the following expression a perfect square:
$x^2+2xy+2y^2$.

Answer 1

${(x+y)}^2=z^2-y^2$

$m(x+y)=n(z+y)$

$n(x+y)=n(z-y)$

By cross multiplication

$\frac{x}{-m^2+2mn+n^2}=\frac{y}{m^2-n^2}=\frac{z}{m^2+n^2}$

Hence, $x=-m^2+2mn+n^2,y=m^2-n^2$ and the hypotenuse is $m^2+n^2$.