Please look at the photo

asked Oct 12, 2023 137k views

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

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