AMC 12 2020a Problem 18

Quadrilateral $ABCD$ satisfies $\angle ABC = \angle ACD = 90^{\circ}, AC = 20$, and $CD = 30$. Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at point $E$, and $AE = 5$. What is the area of quadrilateral $ABCD$?

$\textbf{(A) } 330 \textbf{(B) } 340 \textbf{(C) } 350 \textbf{(D) } 360 \textbf{(E) } 370$