2023 AMC 10A Problem 17

Let ABCD be a rectangle with AB = 30 and BC = 28. Points P and Q lie on BC and CD respectively so that all sides of \triangle ABP, \triangle PCQ, and \triangle QDA have integer lengths. What is the perimeter of \triangle APQ?

\textbf{(A)} 84 \qquad \textbf{(B)} 86 \qquad \textbf{(C)} 88 \qquad \textbf{(D)} 90 \qquad \textbf{(E)} 92

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