1. **Problem statement:** We have two polygons ABCD and EFGH. Polygon EFGH is the image of ABCD by a translation. We know the lengths: AB = 13 cm, BC = $x - 1$ cm, EF = $y + 1$ cm, FG = 8 cm. We need to find the value of $x + y$. 2. **Key concept:** A translation moves every point of a figure the same distance in the same direction. Therefore, corresponding sides of the polygons are equal in length. 3. **Apply the translation property:** Since EFGH is the image of ABCD by translation, corresponding sides are equal: - $AB = EF$ - $BC = FG$ 4. **Set up equations:** From $AB = EF$: $$13 = y + 1$$ From $BC = FG$: $$x - 1 = 8$$ 5. **Solve for $x$ and $y$:** From $13 = y + 1$: $$y = 13 - 1 = 12$$ From $x - 1 = 8$: $$x = 8 + 1 = 9$$ 6. **Find $x + y$:** $$x + y = 9 + 12 = 21$$ **Final answer:** $x + y = 21$