1. **Problem 1: Three externally tangent circles with center distances 7, 8, and 9 inches. Find the area of the largest plate.** 2. The distances between centers of tangent circles equal the sum of their radii. Let the radii be $r_1$, $r_2$, and $r_3$ with $r_3$ largest. 3. We have: $$r_1 + r_2 = 7$$ $$r_2 + r_3 = 8$$ $$r_3 + r_1 = 9$$ 4. Add all three equations: $$2(r_1 + r_2 + r_3) = 7 + 8 + 9 = 24$$ $$r_1 + r_2 + r_3 = 12$$ 5. From $r_1 + r_2 = 7$, substitute into sum: $$7 + r_3 = 12 \implies r_3 = 5$$ 6. The largest radius is $r_3 = 5$ inches. 7. Area of largest plate: $$\pi r_3^2 = \pi \times 5^2 = 25\pi$$ --- 8. **Problem 2: A chord of length 8 units is 3 units from the center of circle C. Find the area of circle C.** 9. Let radius be $r$. The perpendicular distance from center to chord is 3 units. 10. Using right triangle formed by radius, half chord, and distance to chord: $$r^2 = 3^2 + \left(\frac{8}{2}\right)^2 = 9 + 16 = 25$$ 11. Radius: $$r = 5$$ 12. Area of circle: $$\pi r^2 = \pi \times 5^2 = 25\pi$$ **Final answers:** - Largest plate area: $25\pi$ - Circle C area: $25\pi$