To solve the problem of identifying which reflection will produce an image of [tex]\(\triangle RST\)[/tex] with a vertex at [tex]\((2, -3)\)[/tex], let's go through each reflection one-by-one and observe the resulting coordinates of the vertex:
1. Reflection across the x-axis:
- The reflection of a point [tex]\((x, y)\)[/tex] across the x-axis is [tex]\((x, -y)\)[/tex].
- For the vertex [tex]\((2, -3)\)[/tex], reflecting across the x-axis results in [tex]\((2, 3)\)[/tex].
2. Reflection across the y-axis:
- The reflection of a point [tex]\((x, y)\)[/tex] across the y-axis is [tex]\((-x, y)\)[/tex].
- For the vertex [tex]\((2, -3)\)[/tex], reflecting across the y-axis results in [tex]\((-2, -3)\)[/tex].
3. Reflection across the line [tex]\(y = x\)[/tex]:
- The reflection of a point [tex]\((x, y)\)[/tex] across the line [tex]\(y = x\)[/tex] is [tex]\((y, x)\)[/tex].
- For the vertex [tex]\((2, -3)\)[/tex], reflecting across the line [tex]\(y = x\)[/tex] results in [tex]\((-3, 2)\)[/tex].
4. Reflection across the line [tex]\(y = -x\)[/tex]:
- The reflection of a point [tex]\((x, y)\)[/tex] across the line [tex]\(y = -x\)[/tex] is [tex]\((-y, -x)\)[/tex].
- For the vertex [tex]\((2, -3)\)[/tex], reflecting across the line [tex]\(y = -x\)[/tex] results in [tex]\((3, -2)\)[/tex].
Now, comparing the resulting coordinates from each reflection with the given vertex [tex]\((2, -3)\)[/tex], we see that none of the reflections result in the vertex [tex]\((2, -3)\)[/tex]:
- Reflection across the x-axis results in [tex]\((2, 3)\)[/tex].
- Reflection across the y-axis results in [tex]\((-2, -3)\)[/tex].
- Reflection across the line [tex]\(y = x\)[/tex] results in [tex]\((-3, 2)\)[/tex].
- Reflection across the line [tex]\(y = -x\)[/tex] results in [tex]\((3, -2)\)[/tex].
Hence, none of the reflections mentioned will produce an image of [tex]\(\triangle RST\)[/tex] with a vertex at [tex]\((2, -3)\)[/tex].
