Answer :
To determine which reflection will produce an image of the triangle [tex]\(\triangle RST\)[/tex] with a vertex at [tex]\((2, -3)\)[/tex] after reflecting a point at [tex]\((2, -3)\)[/tex], we need to examine the effect of each type of reflection on this point.
### 1. Reflection across the [tex]\(x\)[/tex]-axis:
Reflecting a point [tex]\((x, y)\)[/tex] across the [tex]\(x\)[/tex]-axis results in a new point where the coordinates are [tex]\((x, -y)\)[/tex]. Therefore, if we reflect the point [tex]\((2, -3)\)[/tex] across the [tex]\(x\)[/tex]-axis:
[tex]\[ (x, y) = (2, -3) \implies (x, -y) = (2, 3) \][/tex]
The reflected point is [tex]\((2, 3)\)[/tex].
### 2. Reflection across the [tex]\(y\)[/tex]-axis:
Reflecting a point [tex]\((x, y)\)[/tex] across the [tex]\(y\)[/tex]-axis results in a new point where the coordinates are [tex]\((-x, y)\)[/tex]. Thus, reflecting the point [tex]\((2, -3)\)[/tex] across the [tex]\(y\)[/tex]-axis:
[tex]\[ (x, y) = (2, -3) \implies (-x, y) = (-2, -3) \][/tex]
The reflected point is [tex]\((-2, -3)\)[/tex].
### 3. Reflection across the line [tex]\(y = x\)[/tex]:
Reflecting a point [tex]\((x, y)\)[/tex] across the line [tex]\(y = x\)[/tex] results in exchanging the coordinates, so the new point is [tex]\((y, x)\)[/tex]. Therefore, reflecting the point [tex]\((2, -3)\)[/tex] across the line [tex]\(y = x\)[/tex]:
[tex]\[ (x, y) = (2, -3) \implies (y, x) = (-3, 2) \][/tex]
The reflected point is [tex]\((-3, 2)\)[/tex].
### 4. Reflection across the line [tex]\(y = -x\)[/tex]:
Reflecting a point [tex]\((x, y)\)[/tex] across the line [tex]\(y = -x\)[/tex] results in swapping and negating the coordinates, giving [tex]\((-y, -x)\)[/tex]. Hence, reflecting the point [tex]\((2, -3)\)[/tex] across the line [tex]\(y = -x\)[/tex]:
[tex]\[ (x, y) = (2, -3) \implies (-y, -x) = (3, -2) \][/tex]
The reflected point is [tex]\((3, -2)\)[/tex].
### Summary:
- Reflection across the [tex]\(x\)[/tex]-axis yields: [tex]\((2, 3)\)[/tex]
- Reflection across the [tex]\(y\)[/tex]-axis yields: [tex]\((-2, -3)\)[/tex]
- Reflection across the line [tex]\(y = x\)[/tex] yields: [tex]\((-3, 2)\)[/tex]
- Reflection across the line [tex]\(y = -x\)[/tex] yields: [tex]\((3, -2)\)[/tex]
Given that the vertex at [tex]\((2, -3)\)[/tex] changes to the corresponding points after various reflections:
- The point [tex]\((2, -3)\)[/tex], after reflection across the [tex]\(y\)[/tex]-axis, becomes [tex]\((-2, -3)\)[/tex].
Hence, the reflection that produces an image of [tex]\(\triangle RST\)[/tex] with a vertex at [tex]\((2,-3)\)[/tex] is:
A reflection of [tex]\(\triangle RST\)[/tex] across the [tex]\(y\)[/tex]-axis.
### 1. Reflection across the [tex]\(x\)[/tex]-axis:
Reflecting a point [tex]\((x, y)\)[/tex] across the [tex]\(x\)[/tex]-axis results in a new point where the coordinates are [tex]\((x, -y)\)[/tex]. Therefore, if we reflect the point [tex]\((2, -3)\)[/tex] across the [tex]\(x\)[/tex]-axis:
[tex]\[ (x, y) = (2, -3) \implies (x, -y) = (2, 3) \][/tex]
The reflected point is [tex]\((2, 3)\)[/tex].
### 2. Reflection across the [tex]\(y\)[/tex]-axis:
Reflecting a point [tex]\((x, y)\)[/tex] across the [tex]\(y\)[/tex]-axis results in a new point where the coordinates are [tex]\((-x, y)\)[/tex]. Thus, reflecting the point [tex]\((2, -3)\)[/tex] across the [tex]\(y\)[/tex]-axis:
[tex]\[ (x, y) = (2, -3) \implies (-x, y) = (-2, -3) \][/tex]
The reflected point is [tex]\((-2, -3)\)[/tex].
### 3. Reflection across the line [tex]\(y = x\)[/tex]:
Reflecting a point [tex]\((x, y)\)[/tex] across the line [tex]\(y = x\)[/tex] results in exchanging the coordinates, so the new point is [tex]\((y, x)\)[/tex]. Therefore, reflecting the point [tex]\((2, -3)\)[/tex] across the line [tex]\(y = x\)[/tex]:
[tex]\[ (x, y) = (2, -3) \implies (y, x) = (-3, 2) \][/tex]
The reflected point is [tex]\((-3, 2)\)[/tex].
### 4. Reflection across the line [tex]\(y = -x\)[/tex]:
Reflecting a point [tex]\((x, y)\)[/tex] across the line [tex]\(y = -x\)[/tex] results in swapping and negating the coordinates, giving [tex]\((-y, -x)\)[/tex]. Hence, reflecting the point [tex]\((2, -3)\)[/tex] across the line [tex]\(y = -x\)[/tex]:
[tex]\[ (x, y) = (2, -3) \implies (-y, -x) = (3, -2) \][/tex]
The reflected point is [tex]\((3, -2)\)[/tex].
### Summary:
- Reflection across the [tex]\(x\)[/tex]-axis yields: [tex]\((2, 3)\)[/tex]
- Reflection across the [tex]\(y\)[/tex]-axis yields: [tex]\((-2, -3)\)[/tex]
- Reflection across the line [tex]\(y = x\)[/tex] yields: [tex]\((-3, 2)\)[/tex]
- Reflection across the line [tex]\(y = -x\)[/tex] yields: [tex]\((3, -2)\)[/tex]
Given that the vertex at [tex]\((2, -3)\)[/tex] changes to the corresponding points after various reflections:
- The point [tex]\((2, -3)\)[/tex], after reflection across the [tex]\(y\)[/tex]-axis, becomes [tex]\((-2, -3)\)[/tex].
Hence, the reflection that produces an image of [tex]\(\triangle RST\)[/tex] with a vertex at [tex]\((2,-3)\)[/tex] is:
A reflection of [tex]\(\triangle RST\)[/tex] across the [tex]\(y\)[/tex]-axis.
