1. **Problem statement:** In rectangle ABCD, with right angles at B and C, points B, K, and C lie on one side such that BK = 30 and KC = 10. We need to find the length of diagonal AC. 2. **Formula and rules:** In a rectangle, the diagonal length can be found using the Pythagorean theorem: $$AC = \sqrt{AB^2 + BC^2}$$ where AB and BC are the lengths of the sides adjacent to the right angle. 3. **Intermediate work:** Since BK and KC are parts of side BC, total BC = BK + KC = 30 + 10 = 40. 4. We do not have AB directly, but since ABCD is a rectangle, AB is perpendicular to BC. Without AB length given, we cannot find AC directly. However, if AB is known or assumed, we can proceed. 5. Assuming AB is known or equal to some length $x$, then: $$AC = \sqrt{x^2 + 40^2} = \sqrt{x^2 + 1600}$$ 6. Without AB length, the problem is incomplete. If AB is given or can be found, substitute and calculate AC. **Final answer:** AC = $$\sqrt{AB^2 + 40^2}$$ where AB is the length of side AB.