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.
