1. **State the problem:** We have two similar polygons (rectangles). One has a side length of 4 in., and the other has a side length of 20 in. The area of the larger polygon is given as $100$ in$^2$. We need to find the area of the smaller polygon. 2. **Recall the formula and rules:** For similar polygons, the ratio of their areas is the square of the ratio of their corresponding side lengths. 3. **Set up the ratio of side lengths:** $$\text{Ratio of sides} = \frac{4}{20} = \frac{1}{5}$$ 4. **Set up the ratio of areas:** $$\text{Ratio of areas} = \left(\frac{1}{5}\right)^2 = \frac{1}{25}$$ 5. **Calculate the area of the smaller polygon:** Let $A_s$ be the area of the smaller polygon. $$\frac{A_s}{100} = \frac{1}{25}$$ 6. **Solve for $A_s$:** $$A_s = 100 \times \frac{1}{25}$$ 7. **Simplify:** $$A_s = \cancel{100} \times \frac{1}{\cancel{25}} = 4$$ **Final answer:** The area of the smaller polygon is $4$ in$^2$.