A rectangular piece of paper has corners labeled A, B, C, and D, with BC = 80 and CD = 120. Let M be the midpoint of side AB. The corner labeled A is folded along line MD and the corner labeled B is folded along line MC until the segments AM and MB coincide. Let S denote the point in space where A and B meet. If H is the foot of the perpendicular from S to the original plane of the paper, find HM.