Slicing a Triangle into $2450$ Pieces

Triangle $ABC$ has side lengths $AB = 9$, $BC = 5\sqrt{3}$, and $AC = 12$. Points $P_0 = A, P_1, P_2, \dots, P_{2449}, P_{2450} = B$ lie in this order on segment $AB$, and points $Q_0 = A, Q_1, Q_2, \dots, Q_{2449}, Q_{2450} = C$ lie in this order on segment $AC$. Each of the $2449$ segments $P_kQ_k$ ($k = 1, 2, \dots, 2449$) is parallel to $BC$, and together they divide the triangle into $2450$ regions — one triangle and $2449$ trapezoids — all of equal area. Find the number of segments $P_kQ_k$ whose length is rational.