2021 AMC 12A Problem 24

Semicircle \Gamma has diameter \overline{AB} of length 14. Circle \Omega lies tangent to \overline{AB} at a point P and intersects \Gamma at points Q and R. If QR=3\sqrt3 and \angle QPR=60^\circ, then the area of \triangle PQR equals \tfrac{a\sqrt{b}}{c}, where a and c are relatively prime positive integers, and b is a positive integer not divisible by the square of any prime. What is a+b+c?

\textbf{(A) }110 \qquad \textbf{(B) }114 \qquad \textbf{(C) }118 \qquad \textbf{(D) }122\qquad \textbf{(E) }126

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