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A line segment has endpoints at [tex]$(-1,4)$[/tex] and [tex]$(4,1)$[/tex]. Which reflection will produce an image with endpoints at [tex]$(-4,1)$[/tex] and [tex]$(-1,-4)$[/tex]?

A. A reflection of the line segment across the [tex]$x$[/tex]-axis
B. A reflection of the line segment across the [tex]$y$[/tex]-axis
C. A reflection of the line segment across the line [tex]$y=x$[/tex]
D. A reflection of the line segment across the line [tex]$y=-x$[/tex]

Sagot :

GinnyAnsweridn
To solve the given problem, we need to determine which reflection will transform the endpoints [tex]\((-1, 4)\)[/tex] and [tex]\((4, 1)\)[/tex] of a line segment into [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex]:

1. Reflection across the [tex]\(x\)[/tex]-axis:
- Reflecting [tex]\((-1, 4)\)[/tex] across the [tex]\(x\)[/tex]-axis results in [tex]\((-1, -4)\)[/tex].
- Reflecting [tex]\((4, 1)\)[/tex] across the [tex]\(x\)[/tex]-axis results in [tex]\((4, -1)\)[/tex].
- This does not match the desired endpoints [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex].

2. Reflection across the [tex]\(y\)[/tex]-axis:
- Reflecting [tex]\((-1, 4)\)[/tex] across the [tex]\(y\)[/tex]-axis results in [tex]\((1, 4)\)[/tex].
- Reflecting [tex]\((4, 1)\)[/tex] across the [tex]\(y\)[/tex]-axis results in [tex]\((-4, 1)\)[/tex].
- This also does not match the desired endpoints [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex].

3. Reflection across the line [tex]\(y = x\)[/tex]:
- Reflecting [tex]\((-1, 4)\)[/tex] across the line [tex]\(y = x\)[/tex] results in [tex]\((4, -1)\)[/tex].
- Reflecting [tex]\((4, 1)\)[/tex] across the line [tex]\(y = x\)[/tex] results in [tex]\((1, 4)\)[/tex].
- This transformation does not match the desired endpoints [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex].

4. Reflection across the line [tex]\(y = -x\)[/tex]:
- Reflecting [tex]\((-1, 4)\)[/tex] across the line [tex]\(y = -x\)[/tex] results in [tex]\((-4, -1)\)[/tex].
- Reflecting [tex]\((4, 1)\)[/tex] across the line [tex]\(y = -x\)[/tex] results in [tex]\((-1, -4)\)[/tex].
- This matches the desired endpoints [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex].

Given that reflecting across the line [tex]\(y = -x\)[/tex] correctly transforms [tex]\((-1, 4)\)[/tex] and [tex]\((4, 1)\)[/tex] to [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex], the correct answer is the reflection across the line [tex]\(y = -x\)[/tex].
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