Call the short leg a. The Pythagorean Theorem says:
[tex]a^2 + (8 + 2 a)^2 = 52^2[/tex]
[tex]a^2 + 8^2 + 32 a + 4a^2 = 52^2[/tex]
[tex]5a^2 + 32 a + 8^2-52^2 = 0[/tex]
[tex]5a^2 + 32 a + (8-52)(8+52) = 0[/tex]
[tex]5a^2 + 32 a - (44)(60) = 0[/tex]
[tex](5a + 132)(a - 20) = 0[/tex]
Only the positive root [tex]a=20[/tex] is relevant here.
The other side is [tex]2a+8=48[/tex]
The perimeter is [tex]20 + 48 + 52 = 120[/tex] inches
