Question

what reflection of the parallelogram abcd results in the image abcd? is the image unique using only one reflection? is the image unique using one reflection and some translation? reflect abcd across the line...

Explanation:

First, identify the reflection line that maps parallelogram ABCD to A'B'C'D': this is the vertical line $x=-1$, as each vertex and its image are equidistant from this line (e.g., A(2,6) maps to A'(-4,6); distance from A to $x=-1$ is $|2 - (-1)|=3$, distance from A' to $x=-1$ is $|-4 - (-1)|=3$).

• For uniqueness with one reflection: Yes, the image is unique because a single reflection over a fixed line produces exactly one congruent image.
• For uniqueness with one reflection + one translation: No, the image is not unique. There are infinitely many combinations of a reflection (over different lines) followed by a translation that can map ABCD to A'B'C'D', as different reflection lines paired with corresponding translations can achieve the same final image.

Answer:

1. The reflection that maps ABCD to A'B'C'D' is across the vertical line $x=-1$.
2. Yes, the image is unique using only one reflection.
3. No, the image is not unique using one reflection and one translation.