Two shaded identical rectangular decorative tiles are first placed (one each) at the top and at the base of a door frame for a hobbit's house, as shown in Figure 1. The distance from W to H is 48 inches. Then the same two tiles are rearranged at the top and at the base of the door frame, as shown in Figure 2. The distance from Y to Z is 31 inches. What is the height of the door frame, in inches?
