1. **Problem Statement:** Quadrilateral RSTU is a kite with diagonals US and TR intersecting at Q. Given side TS = 97 and segment TQ = 65, find the length QS. 2. **Properties of a Kite:** In a kite, the diagonals are perpendicular, and one diagonal is bisected by the other. Here, diagonal TR is bisected at Q, so TQ = QR. 3. **Given:** - TS = 97 - TQ = 65 4. **Goal:** Find QS. 5. **Approach:** Since Q is the midpoint of TR, QR = TQ = 65. 6. Triangle TQS is right-angled at Q because diagonals of a kite are perpendicular. 7. Use the Pythagorean theorem in triangle TQS: $$TS^2 = TQ^2 + QS^2$$ 8. Substitute known values: $$97^2 = 65^2 + QS^2$$ 9. Calculate squares: $$9409 = 4225 + QS^2$$ 10. Solve for $QS^2$: $$QS^2 = 9409 - 4225 = 5184$$ 11. Find QS: $$QS = \sqrt{5184} = 72$$ **Final answer:** $$\boxed{72}$$