AHSME 1979 Problem 24

Sides $AB,~ BC$, and $CD$ of (simple*) quadrilateral $ABCD$ have lengths $4,~ 5$, and $20$, respectively. If vertex angles $B$ and $C$ are obtuse and $\sin C = - \cos B =\frac{3}{5}$, then side $AD$ has length

A polygon is called "simple" if it is not self intersecting. $\textbf{(A) }24 \textbf{(B) }24.5 \textbf{(C) }24.6 \textbf{(D) }24.8 \textbf{(E) }25$