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Sagot :
To determine which reflection will produce an image of the triangle [tex]\(\triangle RST\)[/tex] with a vertex at [tex]\((2, -3)\)[/tex], we will reflect this point across different axes and lines. Here are the steps for each reflection:
1. Reflection across the [tex]\(x\)[/tex]-axis:
- When a point [tex]\((x, y)\)[/tex] is reflected across the [tex]\(x\)[/tex]-axis, the [tex]\(y\)[/tex]-coordinate changes its sign, while the [tex]\(x\)[/tex]-coordinate remains the same.
- Thus, for the point [tex]\((2, -3)\)[/tex], the reflection across the [tex]\(x\)[/tex]-axis will be:
[tex]\[ (2, -(-3)) = (2, 3) \][/tex]
2. Reflection across the [tex]\(y\)[/tex]-axis:
- When a point [tex]\((x, y)\)[/tex] is reflected across the [tex]\(y\)[/tex]-axis, the [tex]\(x\)[/tex]-coordinate changes its sign, while the [tex]\(y\)[/tex]-coordinate remains the same.
- Thus, for the point [tex]\((2, -3)\)[/tex], the reflection across the [tex]\(y\)[/tex]-axis will be:
[tex]\[ (-2, -3) \][/tex]
3. Reflection across the line [tex]\(y = x\)[/tex]:
- When a point [tex]\((x, y)\)[/tex] is reflected across the line [tex]\(y = x\)[/tex], the [tex]\(x\)[/tex]- and [tex]\(y\)[/tex]-coordinates swap places.
- Thus, for the point [tex]\((2, -3)\)[/tex], the reflection across the line [tex]\(y = x\)[/tex] will be:
[tex]\[ (-3, 2) \][/tex]
4. Reflection across the line [tex]\(y = -x\)[/tex]:
- When a point [tex]\((x, y)\)[/tex] is reflected across the line [tex]\(y = -x\)[/tex], the [tex]\(x\)[/tex]- and [tex]\(y\)[/tex]-coordinates swap places, and both coordinates change their signs.
- Thus, for the point [tex]\((2, -3)\)[/tex], the reflection across the line [tex]\(y = -x\)[/tex] will be:
[tex]\[ (-(-3), -(2)) = (3, -2) \][/tex]
Summarizing the reflections:
- Reflection across the [tex]\(x\)[/tex]-axis results in: [tex]\((2, 3)\)[/tex]
- Reflection across the [tex]\(y\)[/tex]-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]
Given these results, the reflection that will produce an image of [tex]\(\triangle RST\)[/tex] with a vertex at [tex]\((2, -3)\)[/tex] is the reflection across the [tex]\(y\)[/tex]-axis.
1. Reflection across the [tex]\(x\)[/tex]-axis:
- When a point [tex]\((x, y)\)[/tex] is reflected across the [tex]\(x\)[/tex]-axis, the [tex]\(y\)[/tex]-coordinate changes its sign, while the [tex]\(x\)[/tex]-coordinate remains the same.
- Thus, for the point [tex]\((2, -3)\)[/tex], the reflection across the [tex]\(x\)[/tex]-axis will be:
[tex]\[ (2, -(-3)) = (2, 3) \][/tex]
2. Reflection across the [tex]\(y\)[/tex]-axis:
- When a point [tex]\((x, y)\)[/tex] is reflected across the [tex]\(y\)[/tex]-axis, the [tex]\(x\)[/tex]-coordinate changes its sign, while the [tex]\(y\)[/tex]-coordinate remains the same.
- Thus, for the point [tex]\((2, -3)\)[/tex], the reflection across the [tex]\(y\)[/tex]-axis will be:
[tex]\[ (-2, -3) \][/tex]
3. Reflection across the line [tex]\(y = x\)[/tex]:
- When a point [tex]\((x, y)\)[/tex] is reflected across the line [tex]\(y = x\)[/tex], the [tex]\(x\)[/tex]- and [tex]\(y\)[/tex]-coordinates swap places.
- Thus, for the point [tex]\((2, -3)\)[/tex], the reflection across the line [tex]\(y = x\)[/tex] will be:
[tex]\[ (-3, 2) \][/tex]
4. Reflection across the line [tex]\(y = -x\)[/tex]:
- When a point [tex]\((x, y)\)[/tex] is reflected across the line [tex]\(y = -x\)[/tex], the [tex]\(x\)[/tex]- and [tex]\(y\)[/tex]-coordinates swap places, and both coordinates change their signs.
- Thus, for the point [tex]\((2, -3)\)[/tex], the reflection across the line [tex]\(y = -x\)[/tex] will be:
[tex]\[ (-(-3), -(2)) = (3, -2) \][/tex]
Summarizing the reflections:
- Reflection across the [tex]\(x\)[/tex]-axis results in: [tex]\((2, 3)\)[/tex]
- Reflection across the [tex]\(y\)[/tex]-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]
Given these results, the reflection that will produce an image of [tex]\(\triangle RST\)[/tex] with a vertex at [tex]\((2, -3)\)[/tex] is the reflection across the [tex]\(y\)[/tex]-axis.
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