1. **Problem statement:** We have a polygon ABCDE with given side lengths: AB = 7 cm (horizontal), BC = 8 cm (vertical), CD = 5 cm (horizontal), DE = 2 cm (vertical). We need to find the length of the diagonal AE (the red line). 2. **Understanding the shape:** Since AB and CD are horizontal and BC and DE are vertical, the polygon forms a step-like shape. Points A and E are connected diagonally. 3. **Coordinates assignment:** Place point A at the origin $(0,0)$. - Point B is 7 cm to the right: $(7,0)$. - Point C is 8 cm up from B: $(7,8)$. - Point D is 5 cm to the right from C: $(12,8)$. - Point E is 2 cm down from D: $(12,6)$. 4. **Calculate the length of AE:** Use the distance formula between points A$(0,0)$ and E$(12,6)$: $$AE = \sqrt{(12-0)^2 + (6-0)^2} = \sqrt{12^2 + 6^2} = \sqrt{144 + 36} = \sqrt{180}$$ 5. **Simplify the square root:** $$\sqrt{180} = \sqrt{36 \times 5} = 6\sqrt{5}$$ 6. **Final answer:** The length of the red line AE is $6\sqrt{5}$ cm, approximately $13.42$ cm.