Equal Perimeters in a Sliced Triangle

Triangle $ABC$ has $AB = 7$, $BC = 8$, and $CA = 10$. Points $P$ and $Q$ are on sides $AB$ and $AC$ respectively such that $PQ$ is parallel to $BC$ and the perimeter of triangle $APQ$ equals the perimeter of trapezoid $BPQC$. The length of $PQ$ can be expressed as $\dfrac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.