Triangle A B C has side lengths A B=9, B C=5 \sqrt{3}, and A C=12. Points A=P_{0}, P_{1}, P_{2}, \ldots, P_{2450}=B are on segment \overline{A B} 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 are on segment \overline{A C} with Q_{k} between Q_{k-1} and Q_{k+1} for k=1,2, \ldots, 2449. Furthermore, each segment \overline{P_{k} Q_{k}}, k=1,2, \ldots, 2449, is parallel to \overline{B C}. The segments cut the triangle into 2450 regions, consisting of 2449 trapezoids and 1 triangle. Each of the 2450 regions has the same area. Find the number of segments \overline{P_{k} Q_{k}}, k=1,2, \ldots, 2450, that have rational length.