2018 AIME II

Triangle $ABC$ has sides $AB=9, BC=5\sqrt3$ and $AC=12.$ Points $A=P_0, P_1, P_2, \ldots , P_{2450}=B$ are on the segment $\overline{AB}$ with $P_k$ between $P_{k-1}$ and $P_{k+1}$ for $k=1,2,\ldots , 2449,$ and points $A=Q_0, Q_1, Q_2, \ldots , Q_{2450}=C$ for $k=1,2,\ldots , 2449$ . Furthermore, each segment $\overline{P_kQ_k}, k=1,2, \ldots , 2449,$ is parallel to $\overline{BC}.$ The segments cut the triangle into $2450$ regions, consisting of $2449$ trapezoids and $1$ triangle. Each of the $2450$ regions have the same area. Find the number of segments $\overline{P_kQ_k}, k=1,2, \ldots , 2450,$ that have rational length.