1. **State the problem:** We have a circle with center $O$ and points $A, B, C, D$ on the circumference. The sector $COD$ has an area of 50.24 square units and a central angle of 90°. We need to find the radius of the circle and the measure of angle $AB$ (which is 30° at $O$). 2. **Formula for the area of a sector:** The area $A$ of a sector with radius $r$ and central angle $\theta$ (in degrees) is given by: $$A = \frac{\theta}{360} \pi r^2$$ 3. **Calculate the radius:** Given $A = 50.24$ and $\theta = 90°$, substitute into the formula: $$50.24 = \frac{90}{360} \times 3.14 \times r^2$$ Simplify the fraction: $$50.24 = \frac{1}{4} \times 3.14 \times r^2$$ Multiply both sides by 4: $$4 \times 50.24 = 3.14 \times r^2$$ $$200.96 = 3.14 \times r^2$$ Divide both sides by 3.14: $$\frac{200.96}{3.14} = r^2$$ $$r^2 = 64$$ Take the square root: $$r = \sqrt{64} = 8$$ 4. **Measure of angle $AB$:** Given $m \angle AB = 30°$ (already provided in the problem), so: $$m \angle AB = 30°$$ **Final answers:** - Radius of the circle $r = 8$ units - Measure of angle $AB = 30°$