2006 Cayley Problem 25

A semi-circle of radius 8 cm, rocks back and forth along a line. The distance between the line on which the semi-circle sits and the line above is 12 cm. As it rocks without slipping, the semi-circle touches the line above at two points. (When the semi-circle hits the line above, it immediately rocks back in the other direction.) The distance between these two points, in millimetres, is closest to

\textbf{(A)}\ 55\quad \textbf{(B)}\ 53\quad \textbf{(C)}\ 51\quad \textbf{(D)}\ 49\quad \textbf{(E)}\ 47

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Cayley contests here.