1981 AMC 12 第 19 题

Extend $BN$ through $N$ to meet $AC$

Use congruence across the angle bisector, then apply the midpoint theorem

Extend $BN$ to meet $AC$ at $E.$ The right triangles $ABN$ and $AEN$ are congruent because $AN$ is common and bisects the angle at $A.$ Hence $BE$ is bisected by $N$ and $AE=AB=14.$ Since $AC=19,$ $EC=5.$ In triangle $BEC,$ points $N$ and $M$ are the midpoints of $BE$ and $BC,$ so $MN=\frac{EC}{2}=\frac{5}{2}.$

Therefore, the correct answer is B.