1. **State the problem:** We are given the coordinates of the center of a square at $C(12,7)$ and one vertex at $V(15,10)$. We need to find the area of the square. 2. **Formula and important rules:** The distance from the center to a vertex of a square is half the length of the diagonal. The area of a square is given by $\text{Area} = s^2$, where $s$ is the side length. 3. **Find the distance from center to vertex:** Use the distance formula: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ Substitute $C(12,7)$ and $V(15,10)$: $$d = \sqrt{(15 - 12)^2 + (10 - 7)^2} = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}$$ 4. **Relate distance to diagonal:** The distance $d$ is half the diagonal $D$ of the square, so: $$d = \frac{D}{2} \implies D = 2d = 2 \times 3\sqrt{2} = 6\sqrt{2}$$ 5. **Find the side length $s$:** The diagonal and side length relate by: $$D = s\sqrt{2} \implies s = \frac{D}{\sqrt{2}} = \frac{6\sqrt{2}}{\sqrt{2}} = 6$$ 6. **Calculate the area:** $$\text{Area} = s^2 = 6^2 = 36$$ **Final answer:** The area of the square is $36$ square units.