1. **Problem Statement:** We have a kite-shaped quadrilateral UVWX with diagonals intersecting at point Y. Given that segment YV = 42 and segment WV = 58, we need to find the length WY. 2. **Properties of a Kite:** In a kite, the diagonals intersect at right angles, and one diagonal is bisected by the other. Specifically, the diagonal connecting the vertices where the kite's unequal sides meet is bisected. 3. **Using the Property:** Since UVWX is a kite, diagonal WX is bisected by diagonal UV at point Y. This means that Y is the midpoint of WX, so WY = YV. 4. **Given Lengths:** We know YV = 42 and WV = 58. 5. **Finding WY:** Since Y is the midpoint of WX, WY = YV = 42. **Final Answer:** $$WY = 42$$