1996 AHSME 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 \frac{m}{n}, where m and n are relatively prime positive integers. Find m+n.

\text{(A)}\ 309\qquad\text{(B)}\ 349\qquad\text{(C)}\ 369\qquad\text{(D)}\ 389\qquad\text{(E)}\ 409

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.