Discover how IDNLearn.com can help you learn and grow with its extensive Q&A platform. Get comprehensive and trustworthy answers to all your questions from our knowledgeable community members.
Sagot :
To determine which reflection produces the specified endpoints, let's review various transformations:
1. Reflection across the [tex]\(x\)[/tex]-axis:
- Reflecting a point [tex]\((x, y)\)[/tex] across the [tex]\(x\)[/tex]-axis results in [tex]\((x, -y)\)[/tex].
- For [tex]\((-4, -6)\)[/tex], reflecting it across the [tex]\(x\)[/tex]-axis gives us [tex]\((-4, 6)\)[/tex].
- For [tex]\((-6, 4)\)[/tex], reflecting it across the [tex]\(x\)[/tex]-axis gives us [tex]\((-6, -4)\)[/tex].
- The endpoints after reflection are: [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 [tex]\((-x, y)\)[/tex].
- For [tex]\((-4, -6)\)[/tex], reflecting it across the [tex]\(y\)[/tex]-axis gives us [tex]\((4, -6)\)[/tex].
- For [tex]\((-6, 4)\)[/tex], reflecting it across the [tex]\(y\)[/tex]-axis gives us [tex]\((6, 4)\)[/tex].
- The endpoints after reflection are: [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 [tex]\((y, x)\)[/tex].
- For [tex]\((-4, -6)\)[/tex], reflecting it across the line [tex]\(y = x\)[/tex] gives us [tex]\((-6, -4)\)[/tex].
- For [tex]\((-6, 4)\)[/tex], reflecting it across the line [tex]\(y = x\)[/tex] gives us [tex]\((4, -6)\)[/tex].
- The endpoints after reflection are: [tex]\((-6, -4)\)[/tex] and [tex]\((4, -6)\)[/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 [tex]\((-y, -x)\)[/tex].
- For [tex]\((-4, -6)\)[/tex], reflecting it across the line [tex]\(y = -x\)[/tex] gives us [tex]\((6, 4)\)[/tex].
- For [tex]\((-6, 4)\)[/tex], reflecting it across the line [tex]\(y = -x\)[/tex] gives us [tex]\((-4, -6)\)[/tex].
- The endpoints after reflection are: [tex]\((6, 4)\)[/tex] and [tex]\((-4, -6)\)[/tex].
We want the reflected line segment to have endpoints [tex]\((4, -6)\)[/tex] and [tex]\((8, 4)\)[/tex]. Let's compare this with our results:
- Reflection across the [tex]\(x\)[/tex]-axis gives endpoints [tex]\((-4, 6)\)[/tex] and [tex]\((-6, -4)\)[/tex].
- Reflection across the [tex]\(y\)[/tex]-axis gives endpoints [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
- Reflection across the line [tex]\(y = x\)[/tex] gives endpoints [tex]\((-6, -4)\)[/tex] and [tex]\((4, -6)\)[/tex].
- Reflection across the line [tex]\(y = -x\)[/tex] gives endpoints [tex]\((6, 4)\)[/tex] and [tex]\((-4, -6)\)[/tex].
None of the reflections match the desired endpoints of [tex]\((4, -6)\)[/tex] and [tex]\((8, 4)\)[/tex]. Therefore, it is concluded that the specified reflection does not produce the required endpoints.
1. Reflection across the [tex]\(x\)[/tex]-axis:
- Reflecting a point [tex]\((x, y)\)[/tex] across the [tex]\(x\)[/tex]-axis results in [tex]\((x, -y)\)[/tex].
- For [tex]\((-4, -6)\)[/tex], reflecting it across the [tex]\(x\)[/tex]-axis gives us [tex]\((-4, 6)\)[/tex].
- For [tex]\((-6, 4)\)[/tex], reflecting it across the [tex]\(x\)[/tex]-axis gives us [tex]\((-6, -4)\)[/tex].
- The endpoints after reflection are: [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 [tex]\((-x, y)\)[/tex].
- For [tex]\((-4, -6)\)[/tex], reflecting it across the [tex]\(y\)[/tex]-axis gives us [tex]\((4, -6)\)[/tex].
- For [tex]\((-6, 4)\)[/tex], reflecting it across the [tex]\(y\)[/tex]-axis gives us [tex]\((6, 4)\)[/tex].
- The endpoints after reflection are: [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 [tex]\((y, x)\)[/tex].
- For [tex]\((-4, -6)\)[/tex], reflecting it across the line [tex]\(y = x\)[/tex] gives us [tex]\((-6, -4)\)[/tex].
- For [tex]\((-6, 4)\)[/tex], reflecting it across the line [tex]\(y = x\)[/tex] gives us [tex]\((4, -6)\)[/tex].
- The endpoints after reflection are: [tex]\((-6, -4)\)[/tex] and [tex]\((4, -6)\)[/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 [tex]\((-y, -x)\)[/tex].
- For [tex]\((-4, -6)\)[/tex], reflecting it across the line [tex]\(y = -x\)[/tex] gives us [tex]\((6, 4)\)[/tex].
- For [tex]\((-6, 4)\)[/tex], reflecting it across the line [tex]\(y = -x\)[/tex] gives us [tex]\((-4, -6)\)[/tex].
- The endpoints after reflection are: [tex]\((6, 4)\)[/tex] and [tex]\((-4, -6)\)[/tex].
We want the reflected line segment to have endpoints [tex]\((4, -6)\)[/tex] and [tex]\((8, 4)\)[/tex]. Let's compare this with our results:
- Reflection across the [tex]\(x\)[/tex]-axis gives endpoints [tex]\((-4, 6)\)[/tex] and [tex]\((-6, -4)\)[/tex].
- Reflection across the [tex]\(y\)[/tex]-axis gives endpoints [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
- Reflection across the line [tex]\(y = x\)[/tex] gives endpoints [tex]\((-6, -4)\)[/tex] and [tex]\((4, -6)\)[/tex].
- Reflection across the line [tex]\(y = -x\)[/tex] gives endpoints [tex]\((6, 4)\)[/tex] and [tex]\((-4, -6)\)[/tex].
None of the reflections match the desired endpoints of [tex]\((4, -6)\)[/tex] and [tex]\((8, 4)\)[/tex]. Therefore, it is concluded that the specified reflection does not produce the required endpoints.
Thank you for participating in our discussion. We value every contribution. Keep sharing knowledge and helping others find the answers they need. Let's create a dynamic and informative learning environment together. IDNLearn.com provides the answers you need. Thank you for visiting, and see you next time for more valuable insights.
