AMC 12 2018a Problem 11

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?

draw((0,0)--(4,0)--(4,3)--(0,0));
label("$A$", (0,0), SW);
label("$B$", (4,3), NE);
label("$C$", (4,0), SE);
label("$4$", (2,0), S);
label("$3$", (4,1.5), E);
label("$5$", (2,1.5), NW);
fill(origin--(0,0)--(4,3)--(4,0)--cycle, gray);

$\textbf{(A) } 1+\frac12 \sqrt2 \textbf{(B) } \sqrt3 \textbf{(C) } \frac74 \textbf{(D) } \frac{15}{8} \textbf{(E) } 2$