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 examine each option:
1. Reflection across the [tex]\(x\)[/tex]-axis:
- A point [tex]\((x, y)\)[/tex] reflected across the [tex]\(x\)[/tex]-axis becomes [tex]\((x, -y)\)[/tex].
- For the point [tex]\((0, k)\)[/tex], the reflection across the [tex]\(x\)[/tex]-axis would result in [tex]\((0, -k)\)[/tex].
- Since [tex]\((0, -k) \neq (0, k)\)[/tex] unless [tex]\(k = 0\)[/tex], this is not the correct reflection.
2. Reflection across the [tex]\(y\)[/tex]-axis:
- A point [tex]\((x, y)\)[/tex] reflected across the [tex]\(y\)[/tex]-axis becomes [tex]\((-x, y)\)[/tex].
- For the point [tex]\((0, k)\)[/tex], the reflection across the [tex]\(y\)[/tex]-axis would result in [tex]\((0, k)\)[/tex].
- In this case, [tex]\((0, k) = (0, k)\)[/tex], meaning the point remains unchanged by this reflection.
3. Reflection across the line [tex]\(y = x\)[/tex]:
- A point [tex]\((x, y)\)[/tex] reflected across the line [tex]\(y = x\)[/tex] becomes [tex]\((y, x)\)[/tex].
- For the point [tex]\((0, k)\)[/tex], the reflection across the line [tex]\(y = x\)[/tex] would result in [tex]\((k, 0)\)[/tex].
- Since [tex]\((k, 0) \neq (0, k)\)[/tex] unless [tex]\(k = 0\)[/tex], this is not the correct reflection.
4. Reflection across the line [tex]\(y = -x\)[/tex]:
- A point [tex]\((x, y)\)[/tex] reflected across the line [tex]\(y = -x\)[/tex] becomes [tex]\((-y, -x)\)[/tex].
- For the point [tex]\((0, k)\)[/tex], the reflection across the line [tex]\(y = -x\)[/tex] would result in [tex]\((-k, 0)\)[/tex].
- Since [tex]\((-k, 0) \neq (0, k)\)[/tex] unless [tex]\(k = 0\)[/tex], this is not the correct reflection.
Based on the analysis above, the reflection of the point [tex]\((0, k)\)[/tex] that produces an image at the same coordinates [tex]\((0, k)\)[/tex] is the reflection across the [tex]\(y\)[/tex]-axis.
Thus, the correct answer is:
a reflection of the point across the [tex]\(y\)[/tex]-axis.
