Answer :
Let's analyze the reflection of the vertex [tex]\((2, -3)\)[/tex] across different axes and lines to determine which reflection will produce an image of the vertex at [tex]\((2, -3)\)[/tex].
1. Reflection across the [tex]\(x\)[/tex]-axis:
When you reflect a point across the [tex]\(x\)[/tex]-axis, the [tex]\(y\)[/tex]-coordinate changes its sign while the [tex]\(x\)[/tex]-coordinate remains the same.
- Original vertex: [tex]\((2, -3)\)[/tex]
- After reflection: [tex]\( (2, 3) \)[/tex]
2. Reflection across the [tex]\(y\)[/tex]-axis:
When you reflect a point across the [tex]\(y\)[/tex]-axis, the [tex]\(x\)[/tex]-coordinate changes its sign while the [tex]\(y\)[/tex]-coordinate remains the same.
- Original vertex: [tex]\((2, -3)\)[/tex]
- After reflection: [tex]\((-2, -3)\)[/tex]
3. Reflection across the line [tex]\(y = x\)[/tex]:
When you reflect a point across the line [tex]\(y = x\)[/tex], the [tex]\(x\)[/tex]- and [tex]\(y\)[/tex]-coordinates are swapped.
- Original vertex: [tex]\((2, -3)\)[/tex]
- After reflection: [tex]\((-3, 2)\)[/tex]
4. Reflection across the line [tex]\(y = -x\)[/tex]:
When you reflect a point across the line [tex]\(y = -x\)[/tex], the [tex]\(x\)[/tex]- and [tex]\(y\)[/tex]-coordinates are swapped and their signs are changed.
- Original vertex: [tex]\((2, -3)\)[/tex]
- After reflection: [tex]\((3, -2)\)[/tex]
Now let's compare the transformed coordinates with the original vertex [tex]\((2, -3)\)[/tex]:
- Reflection across the [tex]\(x\)[/tex]-axis gives [tex]\((2, 3)\)[/tex]
- Reflection across the [tex]\(y\)[/tex]-axis gives [tex]\((-2, -3)\)[/tex]
- Reflection across the line [tex]\(y = x\)[/tex] gives [tex]\((-3, 2)\)[/tex]
- Reflection across the line [tex]\(y = -x\)[/tex] gives [tex]\((3, -2)\)[/tex]
None of the reflections produce the original vertex [tex]\((2, -3)\)[/tex]. Therefore, no reflection of [tex]\(\triangle RST\)[/tex] across the given lines will give an image with a vertex at [tex]\((2, -3)\)[/tex].
So, the final answer is that none of the given reflections (across the [tex]\(x\)[/tex]-axis, [tex]\(y\)[/tex]-axis, [tex]\(y = x\)[/tex] line, or [tex]\(y = -x\)[/tex] line) will produce a vertex at [tex]\((2, -3)\)[/tex].
1. Reflection across the [tex]\(x\)[/tex]-axis:
When you reflect a point across the [tex]\(x\)[/tex]-axis, the [tex]\(y\)[/tex]-coordinate changes its sign while the [tex]\(x\)[/tex]-coordinate remains the same.
- Original vertex: [tex]\((2, -3)\)[/tex]
- After reflection: [tex]\( (2, 3) \)[/tex]
2. Reflection across the [tex]\(y\)[/tex]-axis:
When you reflect a point across the [tex]\(y\)[/tex]-axis, the [tex]\(x\)[/tex]-coordinate changes its sign while the [tex]\(y\)[/tex]-coordinate remains the same.
- Original vertex: [tex]\((2, -3)\)[/tex]
- After reflection: [tex]\((-2, -3)\)[/tex]
3. Reflection across the line [tex]\(y = x\)[/tex]:
When you reflect a point across the line [tex]\(y = x\)[/tex], the [tex]\(x\)[/tex]- and [tex]\(y\)[/tex]-coordinates are swapped.
- Original vertex: [tex]\((2, -3)\)[/tex]
- After reflection: [tex]\((-3, 2)\)[/tex]
4. Reflection across the line [tex]\(y = -x\)[/tex]:
When you reflect a point across the line [tex]\(y = -x\)[/tex], the [tex]\(x\)[/tex]- and [tex]\(y\)[/tex]-coordinates are swapped and their signs are changed.
- Original vertex: [tex]\((2, -3)\)[/tex]
- After reflection: [tex]\((3, -2)\)[/tex]
Now let's compare the transformed coordinates with the original vertex [tex]\((2, -3)\)[/tex]:
- Reflection across the [tex]\(x\)[/tex]-axis gives [tex]\((2, 3)\)[/tex]
- Reflection across the [tex]\(y\)[/tex]-axis gives [tex]\((-2, -3)\)[/tex]
- Reflection across the line [tex]\(y = x\)[/tex] gives [tex]\((-3, 2)\)[/tex]
- Reflection across the line [tex]\(y = -x\)[/tex] gives [tex]\((3, -2)\)[/tex]
None of the reflections produce the original vertex [tex]\((2, -3)\)[/tex]. Therefore, no reflection of [tex]\(\triangle RST\)[/tex] across the given lines will give an image with a vertex at [tex]\((2, -3)\)[/tex].
So, the final answer is that none of the given reflections (across the [tex]\(x\)[/tex]-axis, [tex]\(y\)[/tex]-axis, [tex]\(y = x\)[/tex] line, or [tex]\(y = -x\)[/tex] line) will produce a vertex at [tex]\((2, -3)\)[/tex].
