Question: User: x
y
Line segment GH in the center, with G at approximately $(-1, -2)$ and H at approximately $(4, 1)$.What point or points on the following object are invariant when reflecting over the line $x=1$?
$(-1,2)$
$(2,0)$
$(1,-1)$
$(-1,-1)$
1. **State the problem:** We need to find which point or points remain unchanged (invariant) when reflected over the vertical line $x=1$.
2. **Reflection rule:** When reflecting a point $(x,y)$ over the line $x=a$, the reflected point is $(2a - x, y)$.
3. **Apply the reflection to each point:**
- For $(-1,2)$: reflected point is $(2\times1 - (-1), 2) = (2 + 1, 2) = (3, 2)$
- For $(2,0)$: reflected point is $(2\times1 - 2, 0) = (2 - 2, 0) = (0, 0)$
- For $(1,-1)$: reflected point is $(2\times1 - 1, -1) = (2 - 1, -1) = (1, -1)$
- For $(-1,-1)$: reflected point is $(2\times1 - (-1), -1) = (2 + 1, -1) = (3, -1)$
4. **Check which points are invariant:** A point is invariant if it equals its reflection.
- $(-1,2) \neq (3,2)$
- $(2,0) \neq (0,0)$
- $(1,-1) = (1,-1)$ (invariant)
- $(-1,-1) \neq (3,-1)$
5. **Conclusion:** The only point invariant under reflection over $x=1$ is $(1,-1)$.