Answer :
To determine which reflection will produce an image of [tex]$\triangle RST$[/tex] with a vertex at [tex]\((2, -3)\)[/tex], we will systematically analyze each reflection option to see which, if any, will result in the given coordinates for one of the vertices.
1. Reflection across the [tex]\(x\)[/tex]-axis:
- When reflecting across the [tex]\(x\)[/tex]-axis, the [tex]\(y\)[/tex]-coordinate changes its sign, while the [tex]\(x\)[/tex]-coordinate remains the same.
- Reflecting [tex]\((2, -3)\)[/tex] across the [tex]\(x\)[/tex]-axis:
[tex]\[ (x, y) \rightarrow (x, -y) \implies (2, -3) \rightarrow (2, 3) \][/tex]
- This reflection results in the vertex [tex]\((2, 3)\)[/tex].
2. Reflection across the [tex]\(y\)[/tex]-axis:
- When reflecting across the [tex]\(y\)[/tex]-axis, the [tex]\(x\)[/tex]-coordinate changes its sign, while the [tex]\(y\)[/tex]-coordinate remains the same.
- Reflecting [tex]\((2, -3)\)[/tex] across the [tex]\(y\)[/tex]-axis:
[tex]\[ (x, y) \rightarrow (-x, y) \implies (2, -3) \rightarrow (-2, -3) \][/tex]
- This reflection results in the vertex [tex]\((-2, -3)\)[/tex].
3. Reflection across the line [tex]\(y = x\)[/tex]:
- When reflecting across the line [tex]\(y = x\)[/tex], the [tex]\(x\)[/tex]-coordinate and [tex]\(y\)[/tex]-coordinate are swapped.
- Reflecting [tex]\((2, -3)\)[/tex] across the line [tex]\(y = x\)[/tex]:
[tex]\[ (x, y) \rightarrow (y, x) \implies (2, -3) \rightarrow (-3, 2) \][/tex]
- This reflection results in the vertex [tex]\((-3, 2)\)[/tex].
4. Reflection across the line [tex]\(y = -x\)[/tex]:
- When reflecting across the line [tex]\(y = -x\)[/tex], the [tex]\(x\)[/tex]-coordinate and [tex]\(y\)[/tex]-coordinate are swapped and both coordinates are negated.
- Reflecting [tex]\((2, -3)\)[/tex] across the line [tex]\(y = -x\)[/tex]:
[tex]\[ (x, y) \rightarrow (-y, -x) \implies (2, -3) \rightarrow (3, -2) \][/tex]
- This reflection results in the vertex [tex]\((3, -2)\)[/tex].
Based on the results of these reflections:
- Reflecting [tex]\((2, -3)\)[/tex] across the [tex]\(x\)[/tex]-axis results in [tex]\((2, 3)\)[/tex].
- Reflecting [tex]\((2, -3)\)[/tex] across the [tex]\(y\)[/tex]-axis results in [tex]\((-2, -3)\)[/tex].
- Reflecting [tex]\((2, -3)\)[/tex] across the line [tex]\(y = x\)[/tex] results in [tex]\((-3, 2)\)[/tex].
- Reflecting [tex]\((2, -3)\)[/tex] across the line [tex]\(y = -x\)[/tex] results in [tex]\((3, -2)\)[/tex].
None of these reflections results in the vertex [tex]\((2, -3)\)[/tex].
Thus, none of these reflections will produce an image of [tex]\(\triangle RST\)[/tex] with a vertex at [tex]\((2, -3)\)[/tex].
1. Reflection across the [tex]\(x\)[/tex]-axis:
- When reflecting across the [tex]\(x\)[/tex]-axis, the [tex]\(y\)[/tex]-coordinate changes its sign, while the [tex]\(x\)[/tex]-coordinate remains the same.
- Reflecting [tex]\((2, -3)\)[/tex] across the [tex]\(x\)[/tex]-axis:
[tex]\[ (x, y) \rightarrow (x, -y) \implies (2, -3) \rightarrow (2, 3) \][/tex]
- This reflection results in the vertex [tex]\((2, 3)\)[/tex].
2. Reflection across the [tex]\(y\)[/tex]-axis:
- When reflecting across the [tex]\(y\)[/tex]-axis, the [tex]\(x\)[/tex]-coordinate changes its sign, while the [tex]\(y\)[/tex]-coordinate remains the same.
- Reflecting [tex]\((2, -3)\)[/tex] across the [tex]\(y\)[/tex]-axis:
[tex]\[ (x, y) \rightarrow (-x, y) \implies (2, -3) \rightarrow (-2, -3) \][/tex]
- This reflection results in the vertex [tex]\((-2, -3)\)[/tex].
3. Reflection across the line [tex]\(y = x\)[/tex]:
- When reflecting across the line [tex]\(y = x\)[/tex], the [tex]\(x\)[/tex]-coordinate and [tex]\(y\)[/tex]-coordinate are swapped.
- Reflecting [tex]\((2, -3)\)[/tex] across the line [tex]\(y = x\)[/tex]:
[tex]\[ (x, y) \rightarrow (y, x) \implies (2, -3) \rightarrow (-3, 2) \][/tex]
- This reflection results in the vertex [tex]\((-3, 2)\)[/tex].
4. Reflection across the line [tex]\(y = -x\)[/tex]:
- When reflecting across the line [tex]\(y = -x\)[/tex], the [tex]\(x\)[/tex]-coordinate and [tex]\(y\)[/tex]-coordinate are swapped and both coordinates are negated.
- Reflecting [tex]\((2, -3)\)[/tex] across the line [tex]\(y = -x\)[/tex]:
[tex]\[ (x, y) \rightarrow (-y, -x) \implies (2, -3) \rightarrow (3, -2) \][/tex]
- This reflection results in the vertex [tex]\((3, -2)\)[/tex].
Based on the results of these reflections:
- Reflecting [tex]\((2, -3)\)[/tex] across the [tex]\(x\)[/tex]-axis results in [tex]\((2, 3)\)[/tex].
- Reflecting [tex]\((2, -3)\)[/tex] across the [tex]\(y\)[/tex]-axis results in [tex]\((-2, -3)\)[/tex].
- Reflecting [tex]\((2, -3)\)[/tex] across the line [tex]\(y = x\)[/tex] results in [tex]\((-3, 2)\)[/tex].
- Reflecting [tex]\((2, -3)\)[/tex] across the line [tex]\(y = -x\)[/tex] results in [tex]\((3, -2)\)[/tex].
None of these reflections results in the vertex [tex]\((2, -3)\)[/tex].
Thus, none of these reflections will produce an image of [tex]\(\triangle RST\)[/tex] with a vertex at [tex]\((2, -3)\)[/tex].
