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{(8 + 4\sqrt2)/(3 + \sqrt3)}`

What a mess.