1. **Problem statement:** Quadrilateral UVWX is a kite. We are asked to find the length of segment WY. 2. **Important properties of a kite:** - A kite has two pairs of adjacent sides that are equal. - The diagonals of a kite are perpendicular. - One diagonal bisects the other. 3. **Given information:** - Segment XV = 42 - Segment WV = 58 - Y is the intersection of diagonals XV and UW. 4. **Step 1: Identify the diagonals and their properties.** Since UVWX is a kite, diagonals XV and UW intersect at Y, and one diagonal bisects the other. 5. **Step 2: Use the property that one diagonal bisects the other.** Assuming diagonal UW bisects XV at Y, then: $$XY = YV = \frac{XV}{2} = \frac{42}{2} = 21$$ 6. **Step 3: Use the Pythagorean theorem to find WY.** Since diagonals are perpendicular, triangle W Y V is right-angled at Y. We know WV = 58 and YV = 21. Using Pythagoras: $$WY = \sqrt{WV^2 - YV^2} = \sqrt{58^2 - 21^2} = \sqrt{3364 - 441} = \sqrt{2923}$$ 7. **Step 4: Simplify the square root if possible.** $$\sqrt{2923} \approx 54.07$$ **Final answer:** $$WY \approx 54.07$$