1. **Problem Statement:** Find the areas of the sectors formed by $\angle DFE$ given the total area and the central angle. 2. **Formula:** The area of a sector is given by $$\text{Area of sector} = \frac{\theta}{360} \times \pi r^2$$ where $\theta$ is the central angle in degrees and $r$ is the radius. 3. **Step-by-step solutions:** **Problem 9:** - Given total area $A = 100\pi$ in² and angle $60^\circ$. - Using formula: $$\frac{\text{Area}}{100\pi} = \frac{60}{360}$$ Multiply both sides by $100\pi$: $$\text{Area} = \cancel{100\pi} \times \frac{60}{360} = 100\pi \times \frac{1}{6} = \frac{100\pi}{6} = 16.67\pi$$ Calculate numeric value: $$16.67 \times 3.1416 = 52.4 \text{ in}^2$$ **Problem 10:** - Given total area $A = 196\pi$ cm² and angle $104^\circ$. - Using formula: $$\frac{\text{Area}}{196\pi} = \frac{104}{360}$$ Multiply both sides by $196\pi$: $$\text{Area} = 196\pi \times \frac{104}{360} = 196\pi \times 0.2889 = 56.6\pi$$ Calculate numeric value: $$56.6 \times 3.1416 = 177.9 \text{ cm}^2$$ **Problem 11:** - Given total area $A = 784\pi$ m² and angle $137^\circ$. - Using formula: $$\frac{\text{Area}}{784\pi} = \frac{137}{360}$$ Multiply both sides by $784\pi$: $$\text{Area} = 784\pi \times \frac{137}{360} = 784\pi \times 0.3806 = 298.3\pi$$ Calculate numeric value: $$298.3 \times 3.1416 = 937.3 \text{ m}^2$$ **Problem 12:** - Given total area $A = 16\pi$ ft² and angle $75^\circ$. - Using formula: $$\frac{\text{Area}}{16\pi} = \frac{75}{360}$$ Multiply both sides by $16\pi$: $$\text{Area} = 16\pi \times \frac{75}{360} = 16\pi \times 0.2083 = 3.33\pi$$ Calculate numeric value: $$3.33 \times 3.1416 = 10.5 \text{ ft}^2$$ 4. **Summary:** The areas of the sectors are approximately: - Problem 9: 52.4 in² - Problem 10: 177.9 cm² - Problem 11: 937.3 m² - Problem 12: 10.5 ft² --- **Next, Part III: Find the indicated measure given the shaded sector area.** **Problem 13:** Find the area of circle $\odot M$ given sector area $A = 56.87$ cm² and central angle $50^\circ$. 1. Use sector area formula: $$A = \frac{\theta}{360} \times \pi r^2$$ 2. Rearrange to find total area $\pi r^2$: $$\pi r^2 = \frac{360}{\theta} \times A = \frac{360}{50} \times 56.87 = 7.2 \times 56.87 = 409.5 \text{ cm}^2$$ **Problem 14:** Find the radius of $\odot M$ given sector area $A = 12.36$ m² and central angle $89^\circ$. 1. Use sector area formula: $$A = \frac{\theta}{360} \times \pi r^2$$ 2. Rearrange to find $r^2$: $$\pi r^2 = \frac{360}{\theta} \times A = \frac{360}{89} \times 12.36 = 4.0449 \times 12.36 = 50.0$$ 3. Solve for $r$: $$r = \sqrt{\frac{50.0}{\pi}} = \sqrt{15.92} = 3.99 \text{ m}$$ --- **Final answers:** - Problem 9: 52.4 in² - Problem 10: 177.9 cm² - Problem 11: 937.3 m² - Problem 12: 10.5 ft² - Problem 13: Total area of $\odot M$ is 409.5 cm² - Problem 14: Radius of $\odot M$ is approximately 3.99 m