1. **Problem Statement:** Reflect the rectangle with vertices $A(1,2), B(4,5), C(7,2), D(4,-1)$ about the line $y=x+1$. 2. **Formula for Reflection about line $y=x+c$:** The reflection of a point $(x,y)$ about the line $y=x+c$ is given by: $$ (x',y') = (y - c, x + c) $$ This swaps and shifts coordinates accordingly. 3. **Apply the formula to each vertex:** - For $A(1,2)$: $$ (x',y') = (2 - 1, 1 + 1) = (1, 2) $$ - For $B(4,5)$: $$ (x',y') = (5 - 1, 4 + 1) = (4, 5) $$ - For $C(7,2)$: $$ (x',y') = (2 - 1, 7 + 1) = (1, 8) $$ - For $D(4,-1)$: $$ (x',y') = (-1 - 1, 4 + 1) = (-2, 5) $$ 4. **Intermediate check:** Notice that $A$ and $B$ map to themselves, indicating they lie on the line or are symmetric. 5. **Final reflected vertices:** $$ A'(1,2), B'(4,5), C'(1,8), D'(-2,5) $$ 6. **Explanation:** Reflection swaps coordinates relative to the line $y=x+1$ and shifts by $c=1$. This transformation preserves distances and angles, producing a mirror image of the rectangle. **Answer:** The reflected rectangle vertices are $A'(1,2), B'(4,5), C'(1,8), D'(-2,5)$.