1952 AHSME 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} \qquad \textbf{(B) \ }\sqrt{b^2-3} \qquad \textbf{(C) \ }\sqrt{b^2+1}+2 \qquad

\textbf{(D) \ }b^2+5 \qquad \textbf{(E) \ }\sqrt{b^2+3}

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