1983 AHSME Problem 24

How many non-congruent right triangles are there such that the perimeter in \text{cm} and the area in \text{cm}^2 are numerically equal?

\textbf{(A)} \ \text{none} \qquad \textbf{(B)} \ 1 \qquad \textbf{(C)} \ 2 \qquad \textbf{(D)} \ 4 \qquad \textbf{(E)} \ \text{infinitely many}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.