Looking for answers? Westonci.ca is your go-to Q&A platform, offering quick, trustworthy responses from a community of experts. Join our Q&A platform and get accurate answers to all your questions from professionals across multiple disciplines. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
To determine which reflection will produce an image with endpoints at [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex] from the original endpoints [tex]\((-4, -6)\)[/tex] and [tex]\((-6, 4)\)[/tex], let's analyze each option:
1. Reflection across the [tex]\(x\)[/tex]-axis:
- Reflecting a point [tex]\((x, y)\)[/tex] across the [tex]\(x\)[/tex]-axis results in the point [tex]\((x, -y)\)[/tex].
- Applying this to the endpoints:
- [tex]\((-4, -6)\)[/tex] reflected across the [tex]\(x\)[/tex]-axis becomes [tex]\((-4, 6)\)[/tex].
- [tex]\((-6, 4)\)[/tex] reflected across the [tex]\(x\)[/tex]-axis becomes [tex]\((-6, -4)\)[/tex].
- The reflected coordinates [tex]\((-4, 6)\)[/tex] and [tex]\((-6, -4)\)[/tex] do not match the desired endpoints [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
2. Reflection across the [tex]\(y\)[/tex]-axis:
- Reflecting a point [tex]\((x, y)\)[/tex] across the [tex]\(y\)[/tex]-axis results in the point [tex]\((-x, y)\)[/tex].
- Applying this to the endpoints:
- [tex]\((-4, -6)\)[/tex] reflected across the [tex]\(y\)[/tex]-axis becomes [tex]\((4, -6)\)[/tex].
- [tex]\((-6, 4)\)[/tex] reflected across the [tex]\(y\)[/tex]-axis becomes [tex]\((6, 4)\)[/tex].
- The reflected coordinates [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex] do match the desired endpoints [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/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 the point [tex]\((y, x)\)[/tex].
- Applying this to the endpoints:
- [tex]\((-4, -6)\)[/tex] reflected across the line [tex]\(y = x\)[/tex] becomes [tex]\((-6, -4)\)[/tex].
- [tex]\((-6, 4)\)[/tex] reflected across the line [tex]\(y = x\)[/tex] becomes [tex]\((4, -6)\)[/tex].
- The reflected coordinates [tex]\((-6, -4)\)[/tex] and [tex]\((4, -6)\)[/tex] do not match the desired endpoints [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/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 the point [tex]\((-y, -x)\)[/tex].
- Applying this to the endpoints:
- [tex]\((-4, -6)\)[/tex] reflected across the line [tex]\(y = -x\)[/tex] becomes [tex]\((6, 4)\)[/tex].
- [tex]\((-6, 4)\)[/tex] reflected across the line [tex]\(y = -x\)[/tex] becomes [tex]\((-4, -6)\)[/tex].
- The reflected coordinates [tex]\((6, 4)\)[/tex] and [tex]\((-4, -6)\)[/tex] do not match the desired endpoints [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
From this analysis, it is clear that:
- Reflecting the points [tex]\((-4, -6)\)[/tex] and [tex]\((-6, 4)\)[/tex] across the [tex]\(y\)[/tex]-axis produces the desired image points [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
Therefore, the correct reflection is:
- A reflection of the line segment across the [tex]\(y\)[/tex]-axis.
The correct answer is:
```
a reflection of the line segment across the y-axis
```
1. Reflection across the [tex]\(x\)[/tex]-axis:
- Reflecting a point [tex]\((x, y)\)[/tex] across the [tex]\(x\)[/tex]-axis results in the point [tex]\((x, -y)\)[/tex].
- Applying this to the endpoints:
- [tex]\((-4, -6)\)[/tex] reflected across the [tex]\(x\)[/tex]-axis becomes [tex]\((-4, 6)\)[/tex].
- [tex]\((-6, 4)\)[/tex] reflected across the [tex]\(x\)[/tex]-axis becomes [tex]\((-6, -4)\)[/tex].
- The reflected coordinates [tex]\((-4, 6)\)[/tex] and [tex]\((-6, -4)\)[/tex] do not match the desired endpoints [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
2. Reflection across the [tex]\(y\)[/tex]-axis:
- Reflecting a point [tex]\((x, y)\)[/tex] across the [tex]\(y\)[/tex]-axis results in the point [tex]\((-x, y)\)[/tex].
- Applying this to the endpoints:
- [tex]\((-4, -6)\)[/tex] reflected across the [tex]\(y\)[/tex]-axis becomes [tex]\((4, -6)\)[/tex].
- [tex]\((-6, 4)\)[/tex] reflected across the [tex]\(y\)[/tex]-axis becomes [tex]\((6, 4)\)[/tex].
- The reflected coordinates [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex] do match the desired endpoints [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/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 the point [tex]\((y, x)\)[/tex].
- Applying this to the endpoints:
- [tex]\((-4, -6)\)[/tex] reflected across the line [tex]\(y = x\)[/tex] becomes [tex]\((-6, -4)\)[/tex].
- [tex]\((-6, 4)\)[/tex] reflected across the line [tex]\(y = x\)[/tex] becomes [tex]\((4, -6)\)[/tex].
- The reflected coordinates [tex]\((-6, -4)\)[/tex] and [tex]\((4, -6)\)[/tex] do not match the desired endpoints [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/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 the point [tex]\((-y, -x)\)[/tex].
- Applying this to the endpoints:
- [tex]\((-4, -6)\)[/tex] reflected across the line [tex]\(y = -x\)[/tex] becomes [tex]\((6, 4)\)[/tex].
- [tex]\((-6, 4)\)[/tex] reflected across the line [tex]\(y = -x\)[/tex] becomes [tex]\((-4, -6)\)[/tex].
- The reflected coordinates [tex]\((6, 4)\)[/tex] and [tex]\((-4, -6)\)[/tex] do not match the desired endpoints [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
From this analysis, it is clear that:
- Reflecting the points [tex]\((-4, -6)\)[/tex] and [tex]\((-6, 4)\)[/tex] across the [tex]\(y\)[/tex]-axis produces the desired image points [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
Therefore, the correct reflection is:
- A reflection of the line segment across the [tex]\(y\)[/tex]-axis.
The correct answer is:
```
a reflection of the line segment across the y-axis
```
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.
