At Westonci.ca, we connect you with the answers you need, thanks to our active and informed community. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
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], let's analyze each reflection option step-by-step.
Step 1: Reflection across the [tex]\(x\)[/tex]-axis
The reflection of each point [tex]\((x, y)\)[/tex] across the [tex]\(x\)[/tex]-axis is [tex]\((x, -y)\)[/tex].
- For point [tex]\((-4, -6)\)[/tex]:
[tex]\[ (x, y) \rightarrow (x, -y) \Rightarrow (-4, -6) \rightarrow (-4, 6) \][/tex]
- For point [tex]\((-6, 4)\)[/tex]:
[tex]\[ (x, y) \rightarrow (x, -y) \Rightarrow (-6, 4) \rightarrow (-6, -4) \][/tex]
The resulting points are [tex]\((-4, 6)\)[/tex] and [tex]\((-6, -4)\)[/tex], which do not match the target points [tex]\((4, 6)\)[/tex] and [tex]\((6, 4)\)[/tex].
Step 2: Reflection across the [tex]\(y\)[/tex]-axis
The reflection of each point [tex]\((x, y)\)[/tex] across the [tex]\(y\)[/tex]-axis is [tex]\((-x, y)\)[/tex].
- For point [tex]\((-4, -6)\)[/tex]:
[tex]\[ (x, y) \rightarrow (-x, y) \Rightarrow (-4, -6) \rightarrow (4, -6) \][/tex]
- For point [tex]\((-6, 4)\)[/tex]:
[tex]\[ (x, y) \rightarrow (-x, y) \Rightarrow (-6, 4) \rightarrow (6, 4) \][/tex]
The resulting points are [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex], which do not match the target points [tex]\((4, 6)\)[/tex] and [tex]\((6, 4)\)[/tex].
Step 3: Reflection across the line [tex]\(y = x\)[/tex]
The reflection of each point [tex]\((x, y)\)[/tex] across the line [tex]\(y = x\)[/tex] is [tex]\((y, x)\)[/tex].
- For point [tex]\((-4, -6)\)[/tex]:
[tex]\[ (x, y) \rightarrow (y, x) \Rightarrow (-4, -6) \rightarrow (-6, -4) \][/tex]
- For point [tex]\((-6, 4)\)[/tex]:
[tex]\[ (x, y) \rightarrow (y, x) \Rightarrow (-6, 4) \rightarrow (4, -6) \][/tex]
The resulting points are [tex]\((-6, -4)\)[/tex] and [tex]\((4, -6)\)[/tex], which do not match the target points [tex]\((4, 6)\)[/tex] and [tex]\((6, 4)\)[/tex].
Step 4: Reflection across the line [tex]\(y = -x\)[/tex]
The reflection of each point [tex]\((x, y)\)[/tex] across the line [tex]\(y = -x\)[/tex] is [tex]\((-y, -x)\)[/tex].
- For point [tex]\((-4, -6)\)[/tex]:
[tex]\[ (x, y) \rightarrow (-y, -x) \Rightarrow (-4, -6) \rightarrow (6, 4) \][/tex]
- For point [tex]\((-6, 4)\)[/tex]:
[tex]\[ (x, y) \rightarrow (-y, -x) \Rightarrow (-6, 4) \rightarrow (-4, -6) \][/tex]
The resulting points are [tex]\((6, 4)\)[/tex] and [tex]\((-4, -6)\)[/tex], which also do not match the target points [tex]\((4, 6)\)[/tex] and [tex]\((6, 4)\)[/tex].
Given the evaluation of each type of reflection, we can conclude that none of the provided reflections result in endpoints at [tex]\((4, 6)\)[/tex] and [tex]\((6, 4)\)[/tex]. Therefore, the answer is:
No valid reflection found.
Step 1: Reflection across the [tex]\(x\)[/tex]-axis
The reflection of each point [tex]\((x, y)\)[/tex] across the [tex]\(x\)[/tex]-axis is [tex]\((x, -y)\)[/tex].
- For point [tex]\((-4, -6)\)[/tex]:
[tex]\[ (x, y) \rightarrow (x, -y) \Rightarrow (-4, -6) \rightarrow (-4, 6) \][/tex]
- For point [tex]\((-6, 4)\)[/tex]:
[tex]\[ (x, y) \rightarrow (x, -y) \Rightarrow (-6, 4) \rightarrow (-6, -4) \][/tex]
The resulting points are [tex]\((-4, 6)\)[/tex] and [tex]\((-6, -4)\)[/tex], which do not match the target points [tex]\((4, 6)\)[/tex] and [tex]\((6, 4)\)[/tex].
Step 2: Reflection across the [tex]\(y\)[/tex]-axis
The reflection of each point [tex]\((x, y)\)[/tex] across the [tex]\(y\)[/tex]-axis is [tex]\((-x, y)\)[/tex].
- For point [tex]\((-4, -6)\)[/tex]:
[tex]\[ (x, y) \rightarrow (-x, y) \Rightarrow (-4, -6) \rightarrow (4, -6) \][/tex]
- For point [tex]\((-6, 4)\)[/tex]:
[tex]\[ (x, y) \rightarrow (-x, y) \Rightarrow (-6, 4) \rightarrow (6, 4) \][/tex]
The resulting points are [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex], which do not match the target points [tex]\((4, 6)\)[/tex] and [tex]\((6, 4)\)[/tex].
Step 3: Reflection across the line [tex]\(y = x\)[/tex]
The reflection of each point [tex]\((x, y)\)[/tex] across the line [tex]\(y = x\)[/tex] is [tex]\((y, x)\)[/tex].
- For point [tex]\((-4, -6)\)[/tex]:
[tex]\[ (x, y) \rightarrow (y, x) \Rightarrow (-4, -6) \rightarrow (-6, -4) \][/tex]
- For point [tex]\((-6, 4)\)[/tex]:
[tex]\[ (x, y) \rightarrow (y, x) \Rightarrow (-6, 4) \rightarrow (4, -6) \][/tex]
The resulting points are [tex]\((-6, -4)\)[/tex] and [tex]\((4, -6)\)[/tex], which do not match the target points [tex]\((4, 6)\)[/tex] and [tex]\((6, 4)\)[/tex].
Step 4: Reflection across the line [tex]\(y = -x\)[/tex]
The reflection of each point [tex]\((x, y)\)[/tex] across the line [tex]\(y = -x\)[/tex] is [tex]\((-y, -x)\)[/tex].
- For point [tex]\((-4, -6)\)[/tex]:
[tex]\[ (x, y) \rightarrow (-y, -x) \Rightarrow (-4, -6) \rightarrow (6, 4) \][/tex]
- For point [tex]\((-6, 4)\)[/tex]:
[tex]\[ (x, y) \rightarrow (-y, -x) \Rightarrow (-6, 4) \rightarrow (-4, -6) \][/tex]
The resulting points are [tex]\((6, 4)\)[/tex] and [tex]\((-4, -6)\)[/tex], which also do not match the target points [tex]\((4, 6)\)[/tex] and [tex]\((6, 4)\)[/tex].
Given the evaluation of each type of reflection, we can conclude that none of the provided reflections result in endpoints at [tex]\((4, 6)\)[/tex] and [tex]\((6, 4)\)[/tex]. Therefore, the answer is:
No valid reflection found.
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.
