A paper triangle with sides of lengths $3,4,$ and $5$ inches, as shown, is folded so that point $A$ falls on point $B$. What is the length in inches of the crease?
$\textbf{(A) } 1+\frac12 \sqrt2 \qquad$ $ \textbf{(B) } \sqrt3 \qquad$ $ \textbf{(C) } \frac74 \qquad$ $ \textbf{(D) } \frac{15}{8} \qquad$ $ \textbf{(E) } 2$