Westonci.ca is the premier destination for reliable answers to your questions, provided by a community of experts. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
To determine which reflection will transform the endpoints of the line segment from [tex]\((-4, -6)\)[/tex] and [tex]\((-6, 4)\)[/tex] to [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex], we need to consider the effect each type of reflection has on the coordinates of points.
1. Reflection across the x-axis:
For a point [tex]\((x, y)\)[/tex], reflection across the x-axis transforms the point to [tex]\((x, -y)\)[/tex].
- Original endpoints: [tex]\((-4, -6)\)[/tex] and [tex]\((-6, 4)\)[/tex]
- After reflection across the x-axis:
[tex]\[ (-4, -6) \rightarrow (-4, 6) \][/tex]
[tex]\[ (-6, 4) \rightarrow (-6, -4) \][/tex]
These reflected points, [tex]\((-4, 6)\)[/tex] and [tex]\((-6, -4)\)[/tex], do not match [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
2. Reflection across the y-axis:
For a point [tex]\((x, y)\)[/tex], reflection across the y-axis transforms the point to [tex]\((-x, y)\)[/tex].
- Original endpoints: [tex]\((-4, -6)\)[/tex] and [tex]\((-6, 4)\)[/tex]
- After reflection across the y-axis:
[tex]\[ (-4, -6) \rightarrow (4, -6) \][/tex]
[tex]\[ (-6, 4) \rightarrow (6, 4) \][/tex]
These reflected points, [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex], match the desired endpoints.
3. Reflection across the line [tex]\(y = x\)[/tex]:
For a point [tex]\((x, y)\)[/tex], reflection across the line [tex]\(y = x\)[/tex] transforms the point to [tex]\((y, x)\)[/tex].
- Original endpoints: [tex]\((-4, -6)\)[/tex] and [tex]\((-6, 4)\)[/tex]
- After reflection across the line [tex]\(y = x\)[/tex]:
[tex]\[ (-4, -6) \rightarrow (-6, -4) \][/tex]
[tex]\[ (-6, 4) \rightarrow (4, -6) \][/tex]
These reflected points, [tex]\((-6, -4)\)[/tex] and [tex]\((4, -6)\)[/tex], do not match [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
4. Reflection across the line [tex]\(y = -x\)[/tex]:
For a point [tex]\((x, y)\)[/tex], reflection across the line [tex]\(y = -x\)[/tex] transforms the point to [tex]\((-y, -x)\)[/tex].
- Original endpoints: [tex]\((-4, -6)\)[/tex] and [tex]\((-6, 4)\)[/tex]
- After reflection across the line [tex]\(y = -x\)[/tex]:
[tex]\[ (-4, -6) \rightarrow (6, 4) \][/tex]
[tex]\[ (-6, 4) \rightarrow (-4, -6) \][/tex]
These reflected points, [tex]\((6, 4)\)[/tex] and [tex]\((-4, -6)\)[/tex], do not match [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
By considering all the reflections and their effects on the given points, the reflection that produces the endpoints [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex] from the original endpoints [tex]\((-4, -6)\)[/tex] and [tex]\((-6, 4)\)[/tex] is the reflection across the y-axis. Therefore, the correct answer is:
A reflection of the line segment across the y-axis.
1. Reflection across the x-axis:
For a point [tex]\((x, y)\)[/tex], reflection across the x-axis transforms the point to [tex]\((x, -y)\)[/tex].
- Original endpoints: [tex]\((-4, -6)\)[/tex] and [tex]\((-6, 4)\)[/tex]
- After reflection across the x-axis:
[tex]\[ (-4, -6) \rightarrow (-4, 6) \][/tex]
[tex]\[ (-6, 4) \rightarrow (-6, -4) \][/tex]
These reflected points, [tex]\((-4, 6)\)[/tex] and [tex]\((-6, -4)\)[/tex], do not match [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
2. Reflection across the y-axis:
For a point [tex]\((x, y)\)[/tex], reflection across the y-axis transforms the point to [tex]\((-x, y)\)[/tex].
- Original endpoints: [tex]\((-4, -6)\)[/tex] and [tex]\((-6, 4)\)[/tex]
- After reflection across the y-axis:
[tex]\[ (-4, -6) \rightarrow (4, -6) \][/tex]
[tex]\[ (-6, 4) \rightarrow (6, 4) \][/tex]
These reflected points, [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex], match the desired endpoints.
3. Reflection across the line [tex]\(y = x\)[/tex]:
For a point [tex]\((x, y)\)[/tex], reflection across the line [tex]\(y = x\)[/tex] transforms the point to [tex]\((y, x)\)[/tex].
- Original endpoints: [tex]\((-4, -6)\)[/tex] and [tex]\((-6, 4)\)[/tex]
- After reflection across the line [tex]\(y = x\)[/tex]:
[tex]\[ (-4, -6) \rightarrow (-6, -4) \][/tex]
[tex]\[ (-6, 4) \rightarrow (4, -6) \][/tex]
These reflected points, [tex]\((-6, -4)\)[/tex] and [tex]\((4, -6)\)[/tex], do not match [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
4. Reflection across the line [tex]\(y = -x\)[/tex]:
For a point [tex]\((x, y)\)[/tex], reflection across the line [tex]\(y = -x\)[/tex] transforms the point to [tex]\((-y, -x)\)[/tex].
- Original endpoints: [tex]\((-4, -6)\)[/tex] and [tex]\((-6, 4)\)[/tex]
- After reflection across the line [tex]\(y = -x\)[/tex]:
[tex]\[ (-4, -6) \rightarrow (6, 4) \][/tex]
[tex]\[ (-6, 4) \rightarrow (-4, -6) \][/tex]
These reflected points, [tex]\((6, 4)\)[/tex] and [tex]\((-4, -6)\)[/tex], do not match [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
By considering all the reflections and their effects on the given points, the reflection that produces the endpoints [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex] from the original endpoints [tex]\((-4, -6)\)[/tex] and [tex]\((-6, 4)\)[/tex] is the reflection across the y-axis. Therefore, the correct answer is:
A reflection of the line segment across the y-axis.
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.
