We want a 30-60-90 triangle with a perimeter of [tex]8 + 4\sqrt2[/tex]. If the shorter leg has length [tex]x[/tex], we must have
[tex]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}}[/tex]
What a mess.
