A circle has center $G$, and points $M$ and $N$ lie on the circle. Line segments $MH$ and $NH$ are tangent to the circle at points $M$ and $N$, respectively. If the radius of the circle is 50 millimeters, and the perimeter of quadrilateral $GMHN$ is 300 millimeters, what is the length, in millimeters, of segment $MH$?