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 option:
1. Reflection of the point across the [tex]\(x\)[/tex]-axis:
- When a point [tex]\((x, y)\)[/tex] is reflected across the [tex]\(x\)[/tex]-axis, the resulting point will be [tex]\((x, -y)\)[/tex].
- For the point [tex]\((0, k)\)[/tex], reflecting across the [tex]\(x\)[/tex]-axis will produce the point [tex]\((0, -k)\)[/tex].
- This does not match the original coordinates [tex]\((0, k)\)[/tex].
2. Reflection of the point across the [tex]\(y\)[/tex]-axis:
- When a point [tex]\((x, y)\)[/tex] is reflected across the [tex]\(y\)[/tex]-axis, the resulting point will be [tex]\((-x, y)\)[/tex].
- For the point [tex]\((0, k)\)[/tex], reflecting across the [tex]\(y\)[/tex]-axis will produce the point [tex]\((0, k)\)[/tex].
- This matches the original coordinates [tex]\((0, k)\)[/tex].
3. Reflection of the point across the line [tex]\(y=x\)[/tex]:
- When a point [tex]\((x, y)\)[/tex] is reflected across the line [tex]\(y=x\)[/tex], the resulting point will be [tex]\((y, x)\)[/tex].
- For the point [tex]\((0, k)\)[/tex], reflecting across the line [tex]\(y=x\)[/tex] will produce the point [tex]\((k, 0)\)[/tex].
- This does not match the original coordinates [tex]\((0, k)\)[/tex].
4. Reflection of the point across the line [tex]\(y=-x\)[/tex]:
- When a point [tex]\((x, y)\)[/tex] is reflected across the line [tex]\(y=-x\)[/tex], the resulting point will be [tex]\((-y, -x)\)[/tex].
- For the point [tex]\((0, k)\)[/tex], reflecting across the line [tex]\(y=-x\)[/tex] will produce the point [tex]\((-k, 0)\)[/tex].
- This does not match the original coordinates [tex]\((0, k)\)[/tex].
Therefore, the reflection of the point [tex]\((0, k)\)[/tex] across the [tex]\(y\)[/tex]-axis will produce an image at the same coordinates, [tex]\((0, k)\)[/tex].
The correct choice is:
- A reflection of the point across the [tex]\(y\)[/tex]-axis.
