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A point has the coordinates [tex]\((0, k)\)[/tex].

Which reflection of the point will produce an image at the same coordinates, [tex]\((0, k)\)[/tex]?

A. A reflection of the point across the [tex]\(x\)[/tex]-axis
B. A reflection of the point across the [tex]\(y\)[/tex]-axis
C. A reflection of the point across the line [tex]\(y=x\)[/tex]
D. A reflection of the point across the line [tex]\(y=-x\)[/tex]

Sagot :

GinnyAnswer
To determine which reflection will produce an image of the point [tex]\((0, k)\)[/tex] at the same coordinates [tex]\((0, k)\)[/tex], we need to examine how each type of reflection affects the coordinates of the point.

1. Reflection across the [tex]$x$[/tex]-axis:
- The reflection of a point [tex]\((x, y)\)[/tex] across the [tex]$x$[/tex]-axis is [tex]\((x, -y)\)[/tex].
- For the point [tex]\((0, k)\)[/tex], reflecting across the [tex]$x$[/tex]-axis gives us the point [tex]\((0, -k)\)[/tex].

2. Reflection across the [tex]$y$[/tex]-axis:
- The reflection of a point [tex]\((x, y)\)[/tex] across the [tex]$y$[/tex]-axis is [tex]\((-x, y)\)[/tex].
- For the point [tex]\((0, k)\)[/tex], reflecting across the [tex]$y$[/tex]-axis gives us the point [tex]\((0, k)\)[/tex].

3. Reflection across the line [tex]$y = x$[/tex]:
- The reflection of a point [tex]\((x, y)\)[/tex] across the line [tex]$y = x$[/tex] is [tex]\((y, x)\)[/tex].
- For the point [tex]\((0, k)\)[/tex], reflecting across the line [tex]$y = x$[/tex] gives us the point [tex]\((k, 0)\)[/tex].

4. Reflection across the line [tex]$y = -x$[/tex]:
- The reflection of a point [tex]\((x, y)\)[/tex] across the line [tex]$y = -x$[/tex] is [tex]\((-y, -x)\)[/tex].
- For the point [tex]\((0, k)\)[/tex], reflecting across the line [tex]$y = -x$[/tex] gives us the point [tex]\((-k, 0)\)[/tex].

From the analysis above, it is clear that reflecting the point [tex]\((0, k)\)[/tex] across the [tex]$y$[/tex]-axis will produce an image at the same coordinates [tex]\((0, k)\)[/tex]. Therefore, the correct reflection type is:

A reflection of the point across the [tex]$y$[/tex]-axis.
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