Discover a world of knowledge at Westonci.ca, where experts and enthusiasts come together to answer your questions. Our platform connects you with professionals ready to provide precise answers to all your questions in various areas of expertise. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
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.
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.
