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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 find the correct reflection that transforms the endpoints of the line segment from [tex]\((-1, 4)\)[/tex] and [tex]\((4, 1)\)[/tex] to [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex], let’s analyze each given option for reflection:

1. Reflection across the x-axis:
- When reflecting a point across the x-axis, the y-coordinate changes its sign.
- Applying this reflection:
- [tex]\((-1, 4)\)[/tex] becomes [tex]\((-1, -4)\)[/tex]
- [tex]\((4, 1)\)[/tex] becomes [tex]\((4, -1)\)[/tex]
- The new reflected endpoints are [tex]\((-1, -4)\)[/tex] and [tex]\((4, -1)\)[/tex], which do not match what we need.

2. Reflection across the y-axis:
- When reflecting a point across the y-axis, the x-coordinate changes its sign.
- Applying this reflection:
- [tex]\((-1, 4)\)[/tex] becomes [tex]\((1, 4)\)[/tex]
- [tex]\((4, 1)\)[/tex] becomes [tex]\((-4, 1)\)[/tex]
- The new reflected endpoints are [tex]\((1, 4)\)[/tex] and [tex]\((-4, 1)\)[/tex], which again do not match what we need.

3. Reflection across the line [tex]\(y = x\)[/tex]:
- When reflecting a point across the line [tex]\(y = x\)[/tex], the coordinates of the point are swapped.
- Applying this reflection:
- [tex]\((-1, 4)\)[/tex] becomes [tex]\((4, -1)\)[/tex]
- [tex]\((4, 1)\)[/tex] becomes [tex]\((1, 4)\)[/tex]
- The new reflected endpoints are [tex]\((4, -1)\)[/tex] and [tex]\((1, 4)\)[/tex], which still do not match what we need.

4. Reflection across the line [tex]\(y = -x\)[/tex]:
- When reflecting a point across the line [tex]\(y = -x\)[/tex], the coordinates of the point swap places and change signs.
- Applying this reflection:
- [tex]\((-1, 4)\)[/tex] becomes [tex]\((-4, 1)\)[/tex]
- [tex]\((4, 1)\)[/tex] becomes [tex]\((-1, -4)\)[/tex]
- The new reflected endpoints are [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex], which match exactly what we need.

Thus, the transformation that will produce the correct image of endpoints is:

A reflection of the line segment across the line [tex]\(y = -x\)[/tex].
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