1954 AHSME Problem 47

At the midpoint of line segment AB which is p units long, a perpendicular MR is erected with length q units. An arc is described from R with a radius equal to \frac{1}{2}AB, meeting AB at T. Then AT and TB are the roots of:

\begin{array}{l} \textbf{(A)}\ x^2+px+q^2=0\\ \textbf{(B)}\ x^2-px+q^2=0\\ \textbf{(C)}\ x^2+px-q^2=0\\ \textbf{(D)}\ x^2-px-q^2=0\\ \textbf{(E)}\ x^2-px+q=0 \end{array}

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