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A line segment has endpoints at [tex]$(3,2)$[/tex] and [tex]$(2,-3)$[/tex]. Which reflection will produce an image with endpoints at [tex][tex]$(3,-2)$[/tex][/tex] and [tex]$(2,3)$[/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]



Answer :

GinnyAnswer
To determine which reflection will transform the original endpoints [tex]\((3,2)\)[/tex] and [tex]\((2,-3)\)[/tex] into the new endpoints [tex]\((3,-2)\)[/tex] and [tex]\((2,3)\)[/tex], let's analyze each possible reflection:

### Original Endpoints:
- [tex]\((3, 2)\)[/tex]
- [tex]\((2, -3)\)[/tex]

### Reflected Endpoints:
- [tex]\((3, -2)\)[/tex]
- [tex]\((2, 3)\)[/tex]

### Reflection across the x-axis:
- For a reflection across the x-axis, the y-coordinates of each point change sign while the x-coordinates remain the same.
- [tex]\((x, y) \rightarrow (x, -y)\)[/tex]

Applying this transformation:
- [tex]\((3, 2) \rightarrow (3, -2)\)[/tex]
- [tex]\((2, -3) \rightarrow (2, 3)\)[/tex]

This matches the given reflected endpoints exactly.

### Reflection across the y-axis:
- For a reflection across the y-axis, the x-coordinates of each point change sign while the y-coordinates remain the same.
- [tex]\((x, y) \rightarrow (-x, y)\)[/tex]

Applying this transformation:
- [tex]\((3, 2) \rightarrow (-3, 2)\)[/tex]
- [tex]\((2, -3) \rightarrow (-2, -3)\)[/tex]

This does not match the given reflected endpoints.

### Reflection across the line [tex]\( y = x \)[/tex]:
- For a reflection across the line [tex]\( y = x \)[/tex], the x- and y-coordinates of each point are swapped.
- [tex]\((x, y) \rightarrow (y, x)\)[/tex]

Applying this transformation:
- [tex]\((3, 2) \rightarrow (2, 3)\)[/tex]
- [tex]\((2, -3) \rightarrow (-3, 2)\)[/tex]

This does not match the given reflected endpoints.

### Reflection across the line [tex]\( y = -x \)[/tex]:
- For a reflection across the line [tex]\( y = -x \)[/tex], the x- and y-coordinates of each point are swapped, and both signs are changed.
- [tex]\((x, y) \rightarrow (-y, -x)\)[/tex]

Applying this transformation:
- [tex]\((3, 2) \rightarrow (-2, -3)\)[/tex]
- [tex]\((2, -3) \rightarrow (3, -2)\)[/tex]

This does not match the given reflected endpoints.

### Conclusion:
Out of all possible reflections, only the reflection across the x-axis accurately transforms the original endpoints [tex]\((3, 2)\)[/tex] and [tex]\((2, -3)\)[/tex] into the reflected endpoints [tex]\((3, -2)\)[/tex] and [tex]\((2, 3)\)[/tex].

So, the correct reflection is:
- A reflection of the line segment across the x-axis.

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