AMC 12 2014b Problem 9

Convex quadrilateral $ABCD$ has $AB=3$, $BC=4$, $CD=13$, $AD=12$, and $\angle ABC=90^{\circ}$, as shown. What is the area of the quadrilateral?

pair A=(0,0), B=(-3,0), C=(-3,-4), D=(48/5,-36/5);
draw(A--B--C--D--A);
label("$A$",A,N); label("$B$",B,NW); label("$C$",C,SW); label("$D$",D,E);
draw(rightanglemark(A,B,C,25));

\textbf{(A)}\ 30 \textbf{(B)}\ 36 \textbf{(C)}\ 40 \textbf{(D)}\ 48 \textbf{(E)}\ 58.5