Answer :
To determine which reflection of the point \((0, k)\) will produce an image at the same coordinates \((0, k)\), we need to analyze the effect of each type of reflection:
1. Reflection across the \(x\)-axis:
- The reflection of a point \((x, y)\) across the \(x\)-axis results in the point \((x, -y)\).
- For the point \((0, k)\), this reflection would produce the point \((0, -k)\).
- Since \((0, -k) \neq (0, k)\), this reflection does not produce the desired result.
2. Reflection across the \(y\)-axis:
- The reflection of a point \((x, y)\) across the \(y\)-axis results in the point \((-x, y)\).
- For the point \((0, k)\), this reflection would produce the point \((0, k)\).
- Since \((0, k) = (0, k)\), this reflection produces the desired result.
3. Reflection across the line \(y = x\):
- The reflection of a point \((x, y)\) across the line \(y = x\) results in the point \((y, x)\).
- For the point \((0, k)\), this reflection would produce the point \((k, 0)\).
- Since \((k, 0) \neq (0, k)\), this reflection does not produce the desired result.
4. Reflection across the line \(y = -x\):
- The reflection of a point \((x, y)\) across the line \(y = -x\) results in the point \((-y, -x)\).
- For the point \((0, k)\), this reflection would produce the point \((-k, 0)\).
- Since \((-k, 0) \neq (0, k)\), this reflection does not produce the desired result.
After analyzing the effects of all possible reflections, we determine that the reflection across the \(y\)-axis is the only one that results in the point \((0, k)\) being mapped to itself.
Therefore, the right choice is:
```
a reflection of the point across the \(y\)-axis
```
Thus, the correct answer is:
```
a reflection of the point across the \(y\)-axis
```
1. Reflection across the \(x\)-axis:
- The reflection of a point \((x, y)\) across the \(x\)-axis results in the point \((x, -y)\).
- For the point \((0, k)\), this reflection would produce the point \((0, -k)\).
- Since \((0, -k) \neq (0, k)\), this reflection does not produce the desired result.
2. Reflection across the \(y\)-axis:
- The reflection of a point \((x, y)\) across the \(y\)-axis results in the point \((-x, y)\).
- For the point \((0, k)\), this reflection would produce the point \((0, k)\).
- Since \((0, k) = (0, k)\), this reflection produces the desired result.
3. Reflection across the line \(y = x\):
- The reflection of a point \((x, y)\) across the line \(y = x\) results in the point \((y, x)\).
- For the point \((0, k)\), this reflection would produce the point \((k, 0)\).
- Since \((k, 0) \neq (0, k)\), this reflection does not produce the desired result.
4. Reflection across the line \(y = -x\):
- The reflection of a point \((x, y)\) across the line \(y = -x\) results in the point \((-y, -x)\).
- For the point \((0, k)\), this reflection would produce the point \((-k, 0)\).
- Since \((-k, 0) \neq (0, k)\), this reflection does not produce the desired result.
After analyzing the effects of all possible reflections, we determine that the reflection across the \(y\)-axis is the only one that results in the point \((0, k)\) being mapped to itself.
Therefore, the right choice is:
```
a reflection of the point across the \(y\)-axis
```
Thus, the correct answer is:
```
a reflection of the point across the \(y\)-axis
```
