IDNLearn.com makes it easy to get reliable answers from knowledgeable individuals. Get step-by-step guidance for all your technical questions from our dedicated community members.
Sagot :
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].
We appreciate your contributions to this forum. Don't forget to check back for the latest answers. Keep asking, answering, and sharing useful information. Thank you for choosing IDNLearn.com. We’re committed to providing accurate answers, so visit us again soon.
