2009 AMC 12B Problem 13

Triangle ABC has AB = 13 and AC = 15, and the altitude to \overline{BC} has length 12. What is the sum of the two possible values of BC?

\textbf{(A)}\ 15\qquad \textbf{(B)}\ 16\qquad \textbf{(C)}\ 17\qquad \textbf{(D)}\ 18\qquad \textbf{(E)}\ 19

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