Question 390664
From the Pythagorean theorem, {{{x^2 + y^2 = 360000}}}.
From the perimeter condition, x + y + 600 = 1400, or y = 800 - x.
Substitute this into the 1st equation, to get
{{{x^2 + (800 - x)^2 = 360000}}},
<==> {{{2x^2 - 1600x + 280000 = 0}}},
<==> {{{x^2 - 800x + 140000 = 0}}}.
Using the quadratic formula, we get

{{{x = (800 +- 200sqrt(2))/2 = 400 +- 100sqrt(2) }}}.
If {{{x = 400 + 100sqrt(2)}}}, then {{{y = 400 - 100sqrt(2)}}}.
If {{{x = 400 - 100sqrt(2)}}}, then {{{y = 400 + 100sqrt(2)}}}.
Hence there is only 1 pair of values for the legs, and those are {{{ 400 - 100sqrt(2)}}} and {{{400 + 100sqrt(2)}}}.