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To determine which reflection produces a vertex at [tex]\((2, -3)\)[/tex] for the triangle [tex]\(\triangle RST\)[/tex], we have to evaluate the possible reflections of an original vertex.
Let's evaluate each possible reflection:
1. Reflection across the [tex]\(x\)[/tex]-axis:
- For a point [tex]\((a, b)\)[/tex], reflecting across the [tex]\(x\)[/tex]-axis will give us the point [tex]\((a, -b)\)[/tex].
- Given an original vertex [tex]\((2, -3)\)[/tex], reflecting across the [tex]\(x\)[/tex]-axis results in:
[tex]\[ (2, -(-3)) = (2, 3) \][/tex]
- So, this reflection gives us the point [tex]\((2, 3)\)[/tex].
2. Reflection across the [tex]\(y\)[/tex]-axis:
- For a point [tex]\((a, b)\)[/tex], reflecting across the [tex]\(y\)[/tex]-axis will give us the point [tex]\((-a, b)\)[/tex].
- Given an original vertex [tex]\((2, -3)\)[/tex], reflecting across the [tex]\(y\)[/tex]-axis results in:
[tex]\[ (-2, -3) \][/tex]
- So, this reflection gives us the point [tex]\((-2, -3)\)[/tex].
3. Reflection across the line [tex]\(y = x\)[/tex]:
- For a point [tex]\((a, b)\)[/tex], reflecting across the line [tex]\(y = x\)[/tex] will give us the point [tex]\((b, a)\)[/tex].
- Given an original vertex [tex]\((2, -3)\)[/tex], reflecting across the line [tex]\(y = x\)[/tex] results in:
[tex]\[ (-3, 2) \][/tex]
- So, this reflection gives us the point [tex]\((-3, 2)\)[/tex].
4. Reflection across the line [tex]\(y = -x\)[/tex]:
- For a point [tex]\((a, b)\)[/tex], reflecting across the line [tex]\(y = -x\)[/tex] will give us the point [tex]\((-b, -a)\)[/tex].
- Given an original vertex [tex]\((2, -3)\)[/tex], reflecting across the line [tex]\(y = -x\)[/tex] results in:
[tex]\[ (3, -2) \][/tex]
- So, this reflection gives us the point [tex]\((3, -2)\)[/tex].
The original vertex that we want after reflection is [tex]\((2, -3)\)[/tex]. Comparing with the results from different reflections:
- Reflecting across the [tex]\(x\)[/tex]-axis: [tex]\((2, 3)\)[/tex]
- Reflecting across the [tex]\(y\)[/tex]-axis: [tex]\((-2, -3)\)[/tex]
- Reflecting across the line [tex]\(y = x\)[/tex]: [tex]\((-3, 2)\)[/tex]
- Reflecting across the line [tex]\(y = -x\)[/tex]: [tex]\((3, -2)\)[/tex]
None of these reflections produce the desired vertex [tex]\((2, -3)\)[/tex]. Therefore, it appears there is no single reflection across the specified lines or axes that will produce the image of the vertex [tex]\((2, -3)\)[/tex] in [tex]\(\triangle RST\)[/tex].
However, the closest reflection to this transformation is the coordinate point [tex]\((-2, -3)\)[/tex], which is achieved by reflecting the triangle across the [tex]\(y\)[/tex]-axis.
Let's evaluate each possible reflection:
1. Reflection across the [tex]\(x\)[/tex]-axis:
- For a point [tex]\((a, b)\)[/tex], reflecting across the [tex]\(x\)[/tex]-axis will give us the point [tex]\((a, -b)\)[/tex].
- Given an original vertex [tex]\((2, -3)\)[/tex], reflecting across the [tex]\(x\)[/tex]-axis results in:
[tex]\[ (2, -(-3)) = (2, 3) \][/tex]
- So, this reflection gives us the point [tex]\((2, 3)\)[/tex].
2. Reflection across the [tex]\(y\)[/tex]-axis:
- For a point [tex]\((a, b)\)[/tex], reflecting across the [tex]\(y\)[/tex]-axis will give us the point [tex]\((-a, b)\)[/tex].
- Given an original vertex [tex]\((2, -3)\)[/tex], reflecting across the [tex]\(y\)[/tex]-axis results in:
[tex]\[ (-2, -3) \][/tex]
- So, this reflection gives us the point [tex]\((-2, -3)\)[/tex].
3. Reflection across the line [tex]\(y = x\)[/tex]:
- For a point [tex]\((a, b)\)[/tex], reflecting across the line [tex]\(y = x\)[/tex] will give us the point [tex]\((b, a)\)[/tex].
- Given an original vertex [tex]\((2, -3)\)[/tex], reflecting across the line [tex]\(y = x\)[/tex] results in:
[tex]\[ (-3, 2) \][/tex]
- So, this reflection gives us the point [tex]\((-3, 2)\)[/tex].
4. Reflection across the line [tex]\(y = -x\)[/tex]:
- For a point [tex]\((a, b)\)[/tex], reflecting across the line [tex]\(y = -x\)[/tex] will give us the point [tex]\((-b, -a)\)[/tex].
- Given an original vertex [tex]\((2, -3)\)[/tex], reflecting across the line [tex]\(y = -x\)[/tex] results in:
[tex]\[ (3, -2) \][/tex]
- So, this reflection gives us the point [tex]\((3, -2)\)[/tex].
The original vertex that we want after reflection is [tex]\((2, -3)\)[/tex]. Comparing with the results from different reflections:
- Reflecting across the [tex]\(x\)[/tex]-axis: [tex]\((2, 3)\)[/tex]
- Reflecting across the [tex]\(y\)[/tex]-axis: [tex]\((-2, -3)\)[/tex]
- Reflecting across the line [tex]\(y = x\)[/tex]: [tex]\((-3, 2)\)[/tex]
- Reflecting across the line [tex]\(y = -x\)[/tex]: [tex]\((3, -2)\)[/tex]
None of these reflections produce the desired vertex [tex]\((2, -3)\)[/tex]. Therefore, it appears there is no single reflection across the specified lines or axes that will produce the image of the vertex [tex]\((2, -3)\)[/tex] in [tex]\(\triangle RST\)[/tex].
However, the closest reflection to this transformation is the coordinate point [tex]\((-2, -3)\)[/tex], which is achieved by reflecting the triangle across the [tex]\(y\)[/tex]-axis.
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