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 what happens to the point under each type of reflection.
1. Reflection across the x-axis:
- Reflecting a point [tex]\((x, y)\)[/tex] across the x-axis results in the point [tex]\((x, -y)\)[/tex].
- For [tex]\((0, k)\)[/tex], the point becomes [tex]\((0, -k)\)[/tex] after reflection.
2. Reflection across the y-axis:
- Reflecting a point [tex]\((x, y)\)[/tex] across the y-axis results in the point [tex]\((-x, y)\)[/tex].
- For [tex]\((0, k)\)[/tex], the point remains [tex]\((0, k)\)[/tex] after reflection.
3. Reflection across the line [tex]\(y = x\)[/tex]:
- Reflecting a point [tex]\((x, y)\)[/tex] across the line [tex]\(y = x\)[/tex] results in the point [tex]\((y, x)\)[/tex].
- For [tex]\((0, k)\)[/tex], the point becomes [tex]\((k, 0)\)[/tex] after reflection.
4. Reflection across the line [tex]\(y = -x\)[/tex]:
- Reflecting a point [tex]\((x, y)\)[/tex] across the line [tex]\(y = -x\)[/tex] results in the point [tex]\((-y, -x)\)[/tex].
- For [tex]\((0, k)\)[/tex], the point becomes [tex]\((-k, 0)\)[/tex] after reflection.
Now, among all these reflections, only the reflection across the y-axis keeps the point’s coordinates the same, which means that the point [tex]\((0, k)\)[/tex] remains [tex]\((0, k)\)[/tex].
Therefore, the correct answer is:
- A reflection of the point across the y-axis will produce an image at the same coordinates [tex]\((0, k)\)[/tex].
