A right triangle has perimeter 32 and area 20. What is the length of its hypotenuse? $\mathrm{(A)}\ \frac{57}{4}\qquad$ $\mathrm{(B)}\ \frac{59}{4}\qquad$ $\mathrm{(C)}\ \frac{61}{4}\qquad$ $\mathrm{(D)}\ \frac{63}{4}\qquad$ $\mathrm{(E)}\ \frac{65}{4}$
Solution
Assume $a, b$ are the legs of the right triangle.
$a+b+\sqrt{a^2+b^2}=32$, or $a^2+b^2=1024-64(a+b)+(a+b)^2$, or $a+b=\frac{2ab+1024}{64}$.
