AHSME 1952 Problem 22

On hypotenuse $AB$ of a right triangle $ABC$ a second right triangle $ABD$ is constructed with hypotenuse $AB$. If $\overline{BC}=1$, $\overline{AC}=b$, and $\overline{AD}=2$, then $\overline{BD}$ equals: $\textbf{(A) \ }\sqrt{b^2+1} \textbf{(B) \ }\sqrt{b^2-3} \textbf{(C) \ }\sqrt{b^2+1}+2$ $\textbf{(D) \ }b^2+5 \textbf{(E) \ }\sqrt{b^2+3}$