To determine which reflection will produce an image of [tex]\(\triangle R S T\)[/tex] with a vertex at [tex]\((2, -3)\)[/tex], we need to analyze how each of the given reflections affects the coordinates of a point.
1. Reflection across the [tex]\(x\)[/tex]-axis:
- This reflection changes a point [tex]\((x, y)\)[/tex] to [tex]\((x, -y)\)[/tex].
- For the vertex [tex]\((2, -3)\)[/tex]:
- Reflecting across the [tex]\(x\)[/tex]-axis would give [tex]\((2, -(-3)) = (2, 3)\)[/tex].
- Therefore, this vertex [tex]\((2, -3)\)[/tex] is not produced by a reflection across the [tex]\(x\)[/tex]-axis.
2. Reflection across the [tex]\(y\)[/tex]-axis:
- This reflection changes a point [tex]\((x, y)\)[/tex] to [tex]\((-x, y)\)[/tex].
- For the vertex [tex]\((2, -3)\)[/tex]:
- Reflecting across the [tex]\(y\)[/tex]-axis would give [tex]\((-2, -3)\)[/tex].
- Therefore, this vertex [tex]\((2, -3)\)[/tex] is not produced by a reflection across the [tex]\(y\)[/tex]-axis.
3. Reflection across the line [tex]\(y = x\)[/tex]:
- This reflection changes a point [tex]\((x, y)\)[/tex] to [tex]\((y, x)\)[/tex].
- For the vertex [tex]\((2, -3)\)[/tex]:
- Reflecting across the line [tex]\(y = x\)[/tex] would give [tex]\((-3, 2)\)[/tex].
- Therefore, this vertex [tex]\((2, -3)\)[/tex] is not produced by a reflection across the line [tex]\(y = x\)[/tex].
4. Reflection across the line [tex]\(y = -x\)[/tex]:
- This reflection changes a point [tex]\((x, y)\)[/tex] to [tex]\((-y, -x)\)[/tex].
- For the vertex [tex]\((2, -3)\)[/tex]:
- Reflecting across the line [tex]\(y = -x\)[/tex] would give [tex]\((-(-3), -(2)) = (3, -2)\)[/tex].
- Therefore, this vertex [tex]\((2, -3)\)[/tex] is not produced by a reflection across the line [tex]\(y = -x\)[/tex].
From these reflections, it is evident that none of the above transformations produce the required vertex at [tex]\((2, -3)\)[/tex].
