AHSME 1952 Problem 24

In the figure, it is given that angle $C = 90^{\circ}$, $\overline{AD} = \overline{DB}$, $DE \perp AB$, $\overline{AB} = 20$, and $\overline{AC} = 12$. The area of quadrilateral $ADEC$ is:

$\textbf{(A)}\ 75 \textbf{(B)}\ 58\frac{1}{2} \textbf{(C)}\ 48 \textbf{(D)}\ 37\frac{1}{2} \textbf{(E)}\ \text{none of these}$