Question: The rectangle ABCD with vertices A($-2$, $6$), B($-6$, $6$), C($-6$, $2$), D($-2$, $2$) is reflected across the mirror line $x = 1$. Determine the coordinates of the image A'B'C'D'.
1. **State the problem:** We have a rectangle ABCD with vertices A($-2$, $6$), B($-6$, $6$), C($-6$, $2$), and D($-2$, $2$). We want to find the coordinates of the reflected image A'B'C'D' across the vertical mirror line $x=1$.
2. **Reflection rule:** When reflecting a point $(x,y)$ across the vertical line $x = k$, the reflected point $(x', y')$ has coordinates:
$$x' = 2k - x, \quad y' = y$$
This means the $x$-coordinate is mirrored about $k$, and the $y$-coordinate remains the same.
3. **Apply the formula to each vertex:**
- For $A(-2,6)$:
$$x' = 2(1) - (-2) = 2 + 2 = 4, \quad y' = 6$$
So, $A' = (4, 6)$.
- For $B(-6,6)$:
$$x' = 2(1) - (-6) = 2 + 6 = 8, \quad y' = 6$$
So, $B' = (8, 6)$.
- For $C(-6,2)$:
$$x' = 2(1) - (-6) = 8, \quad y' = 2$$
So, $C' = (8, 2)$.
- For $D(-2,2)$:
$$x' = 2(1) - (-2) = 4, \quad y' = 2$$
So, $D' = (4, 2)$.
4. **Final answer:** The reflected coordinates are:
$$A'(4, 6), B'(8, 6), C'(8, 2), D'(4, 2)$$
This matches the last option given in the problem.