Question 16
Equilateral $\triangle ABC$ with side length $14$ is rotated about its center by angle $\theta$, where $0 \lt \theta \le 60^\circ$, to form $\triangle DEF$. See the figure. The area of hexagon $ADBECF$ is $91\sqrt{3}$. What is $\tan \theta$?
(A) $\frac{3}{5}$
(B) $\frac{5\sqrt{3}}{11}$
(C) $\frac{4}{5}$
(D) $\frac{11}{13}$
(E) $\frac{7\sqrt{3}}{13}$
(A) $\frac{3}{5}$
(B) $\frac{5\sqrt{3}}{11}$
(C) $\frac{4}{5}$
(D) $\frac{11}{13}$
(E) $\frac{7\sqrt{3}}{13}$