To determine which reflection will produce an image of [tex]\(\triangle RST\)[/tex] with a vertex at [tex]\((2, -3)\)[/tex], we need to consider how each type of reflection transforms the coordinates of a point.
1. Reflection across the x-axis:
- When a point [tex]\((x, y)\)[/tex] is reflected across the x-axis, its new coordinates become [tex]\((x, -y)\)[/tex].
- Applying this transformation to the point [tex]\((2, -3)\)[/tex] results in [tex]\((2, -(-3)) = (2, 3)\)[/tex].
2. Reflection across the y-axis:
- When a point [tex]\((x, y)\)[/tex] is reflected across the y-axis, its new coordinates become [tex]\((-x, y)\)[/tex].
- Applying this transformation to the point [tex]\((2, -3)\)[/tex] results in [tex]\((-2, -3)\)[/tex].
3. Reflection across the line [tex]\(y = x\)[/tex]:
- When a point [tex]\((x, y)\)[/tex] is reflected across the line [tex]\(y = x\)[/tex], its new coordinates become [tex]\((y, x)\)[/tex].
- Applying this transformation to the point [tex]\((2, -3)\)[/tex] results in [tex]\((-3, 2)\)[/tex].
4. Reflection across the line [tex]\(y = -x\)[/tex]:
- When a point [tex]\((x, y)\)[/tex] is reflected across the line [tex]\(y = -x\)[/tex], its new coordinates become [tex]\((-y, -x)\)[/tex].
- Applying this transformation to the point [tex]\((2, -3)\)[/tex] results in [tex]\((-(-3), -2) = (3, -2)\)[/tex].
We need to check if any of the transformations produce the original vertex at [tex]\((2, -3)\)[/tex]:
- For reflection across the x-axis: [tex]\((2, 3)\)[/tex] does not match [tex]\((2, -3)\)[/tex].
- For reflection across the y-axis: [tex]\((-2, -3)\)[/tex] does not match [tex]\((2, -3)\)[/tex].
- For reflection across the line [tex]\(y = x\)[/tex]: [tex]\((-3, 2)\)[/tex] does not match [tex]\((2, -3)\)[/tex].
- For reflection across the line [tex]\(y = -x\)[/tex]: [tex]\((3, -2)\)[/tex] does not match [tex]\((2, -3)\)[/tex].
None of the given reflections will transform a vertex of [tex]\(\triangle RST\)[/tex] into the point [tex]\((2, -3)\)[/tex].
Therefore, there is no reflection among the given options that produces a vertex of [tex]\(\triangle RST\)[/tex] at [tex]\((2, -3)\)[/tex].
So, the correct answer is:
```
None
```
