To determine which reflection of the point [tex]\((0, k)\)[/tex] will produce an image at the same coordinates [tex]\((0, k)\)[/tex], let's analyze each possible reflection:
1. Reflection across the [tex]\(x\)[/tex]-axis:
- The reflection of a point [tex]\((x, y)\)[/tex] across the [tex]\(x\)[/tex]-axis changes its coordinates to [tex]\((x, -y)\)[/tex].
- If we reflect the point [tex]\((0, k)\)[/tex] across the [tex]\(x\)[/tex]-axis, it becomes [tex]\((0, -k)\)[/tex].
2. Reflection across the [tex]\(y\)[/tex]-axis:
- The reflection of a point [tex]\((x, y)\)[/tex] across the [tex]\(y\)[/tex]-axis changes its coordinates to [tex]\((-x, y)\)[/tex].
- If we reflect the point [tex]\((0, k)\)[/tex] across the [tex]\(y\)[/tex]-axis, it becomes [tex]\((0, k)\)[/tex].
3. Reflection across the line [tex]\(y=x\)[/tex]:
- The reflection of a point [tex]\((x, y)\)[/tex] across the line [tex]\(y=x\)[/tex] changes its coordinates to [tex]\((y, x)\)[/tex].
- If we reflect the point [tex]\((0, k)\)[/tex] across the line [tex]\(y=x\)[/tex], it becomes [tex]\((k, 0)\)[/tex].
4. Reflection across the line [tex]\(y=-x\)[/tex]:
- The reflection of a point [tex]\((x, y)\)[/tex] across the line [tex]\(y=-x\)[/tex] changes its coordinates to [tex]\((-y, -x)\)[/tex].
- If we reflect the point [tex]\((0, k)\)[/tex] across the line [tex]\(y=-x\)[/tex], it becomes [tex]\((-k, 0)\)[/tex].
From the above analysis, we see that among all the reflections, only the reflection across the [tex]\(x\)[/tex]-axis keeps the coordinates of the point [tex]\((0, k)\)[/tex] unchanged.
Therefore, the correct answer is:
- a reflection of the point across the [tex]\(x\)[/tex]-axis
