Question

Please look at the photo

Answer 1

We want a 30-60-90 triangle with a perimeter of $8 + 4\sqrt2$. If the shorter leg has length $x$, we must have

$x + x\sqrt3 + 2x = 8 + 4\sqrt2\\3x + x\sqrt3 = 8 + 4\sqrt2\\x(3 + \sqrt3) = 8 + 4\sqrt2\\x = \boxed{\frac{8 + 4\sqrt2}{3 + \sqrt3}}$

What a mess.