1967 AHSME Problem 13

A triangle ABC is to be constructed given a side a (oppisite angle A). angle B, and h_c, the altitude from C. If N is the number of noncongruent solutions, then N

\textbf{(A)}\ \text{is} \; 1\qquad \textbf{(B)}\ \text{is} \; 2\qquad \textbf{(C)}\ \text{must be zero}\qquad \textbf{(D)}\ \text{must be infinite}\qquad \textbf{(E)}\ \text{must be zero or infinite}

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