1. **State the problem:** We need to find various lengths and areas related to two circles: one centered at A with radius 24, and one centered at J with diameter 30. Given lengths and areas correspond to arcs, sectors, and segments. 2. **Formulas and rules:** - Circumference of circle: $$C = 2\pi r$$ - Arc length: $$L = \frac{\theta}{360} \times 2\pi r$$ where $\theta$ is the central angle in degrees. - Area of sector: $$A_{sector} = \frac{\theta}{360} \times \pi r^2$$ - Area of segment: $$A_{segment} = A_{sector} - A_{triangle}$$ where $A_{triangle}$ is the area of the triangle formed by the two radii and the chord. 3. **Circle A (radius 24):** - Circumference: $$C = 2\pi \times 24 = 48\pi$$ - Length of FD = 15 (given) - Length of CFD = 16 (given) - Area of sector BC = 17 (given) - Area of sector DBF = 18 (given) - Area of segment DAE = 19 (given) Since these are given, no calculation needed for these values. 4. **Circle J (diameter 30):** - Radius: $$r = \frac{30}{2} = 15$$ - Circumference: $$C = 2\pi \times 15 = 30\pi$$ - Length of QL = 20 (given) - Length of PNK = 21 (given) - Area of sector NL = 22 (given) - Area of sector QNL = 23 (given) - Area of segment KQ = 24 (given) Again, these are given values. 5. **Summary:** The problem states to find each measure but provides the values directly. If the problem intended to calculate these from angles or other data, that data is not fully provided here. Therefore, the given values are the answers. **Final answers:** - Circle A: Radius = 24 - Length FD = 15 - Length CFD = 16 - Area sector BC = 17 - Area sector DBF = 18 - Area segment DAE = 19 - Circle J: Diameter = 30 - Length QL = 20 - Length PNK = 21 - Area sector NL = 22 - Area sector QNL = 23 - Area segment KQ = 24 These are rounded to the nearest tenth as given.