AHSME 1996 Problem 30

A hexagon inscribed in a circle has three consecutive sides each of length 3 and three consecutive sides each of length 5. The chord of the circle that divides the hexagon into two trapezoids, one with three sides each of length 3 and the other with three sides each of length 5, has length equal to $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. $\textbf{(A)}\ 309 \textbf{(B)}\ 349 \textbf{(C)}\ 369 \textbf{(D)}\ 389 \textbf{(E)}\ 409$