AHSME 1952 Problem 39

If the perimeter of a rectangle is $p$ and its diagonal is $d$, the difference between the length and width of the rectangle is: $\textbf{(A)}\ \frac {\sqrt {8d^2 - p^2}}{2} \textbf{(B)}\ \frac {\sqrt {8d^2 + p^2}}{2} \textbf{(C)}\ \frac{\sqrt{6d^2-p^2}}{2} \\ \textbf{(D)}\ \frac {\sqrt {6d^2 + p^2}}{2} \textbf{(E)}\ \frac {8d^2 - p^2}{4}$