1. **Problem statement:** We want to express the side length $s$ of a square as a function of its diagonal length $d$. Then, express the area $A$ of the square as a function of $d$. 2. **Formula for diagonal of a square:** The diagonal $d$ relates to the side length $s$ by the Pythagorean theorem: $$d = \sqrt{s^2 + s^2} = \sqrt{2s^2} = s\sqrt{2}$$ 3. **Express side length as a function of diagonal:** Solve for $s$: $$s = \frac{d}{\sqrt{2}}$$ 4. **Formula for area of a square:** The area $A$ is: $$A = s^2$$ 5. **Express area as a function of diagonal:** Substitute $s = \frac{d}{\sqrt{2}}$: $$A = \left(\frac{d}{\sqrt{2}}\right)^2 = \frac{d^2}{2}$$ **Final answers:** - Side length as a function of diagonal: $s(d) = \frac{d}{\sqrt{2}}$ - Area as a function of diagonal: $A(d) = \frac{d^2}{2}$