In \(\triangle PQR\), \(\angle PRQ = 90^{\circ}\). An altitude is drawn in \(\triangle PQR\) from \(R\) to \(PQ\), intersecting \(PQ\) at \(S\). A median is drawn in \(\triangle PSR\) from \(P\) to \(SR\), intersecting \(SR\) at \(T\).
If the length of the median \(PT\) is \(39\) and the length of \(PS\) is \(36\), determine the length of \(QS\).
Note: An altitude of a triangle is a line segment drawn from a vertex of the triangle perpendicular to the opposite side. A median is a line segment drawn from a vertex of the triangle to the midpoint of the opposite side.