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Which reflection will produce an image of [tex][tex]$\triangle R S T$[/tex][/tex] with a vertex at [tex][tex]$(2,-3)$[/tex][/tex]?

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

Sagot :

GinnyAnswer
To determine the resulting coordinates when a vertex of a triangle at [tex]\( (2, -3) \)[/tex] is reflected across different lines, we need to understand the effect of each type of reflection.

1. Reflection across the [tex]\( x \)[/tex]-axis:
- When you reflect a point across the [tex]\( x \)[/tex]-axis, the [tex]\( x \)[/tex]-coordinate remains the same, but the [tex]\( y \)[/tex]-coordinate changes sign.
- The coordinates of the vertex [tex]\( (2, -3) \)[/tex] reflected across the [tex]\( x \)[/tex]-axis will be [tex]\( (2, 3) \)[/tex].

2. Reflection across the [tex]\( y \)[/tex]-axis:
- When reflecting a point across the [tex]\( y \)[/tex]-axis, the [tex]\( y \)[/tex]-coordinate remains the same, but the [tex]\( x \)[/tex]-coordinate changes sign.
- The coordinates of the vertex [tex]\( (2, -3) \)[/tex] reflected across the [tex]\( y \)[/tex]-axis will be [tex]\( (-2, -3) \)[/tex].

3. Reflection across the line [tex]\( y = x \)[/tex]:
- Reflecting a point across the line [tex]\( y = x \)[/tex] swaps the [tex]\( x \)[/tex]-coordinate and the [tex]\( y \)[/tex]-coordinate.
- The coordinates of the vertex [tex]\( (2, -3) \)[/tex] reflected across the line [tex]\( y = x \)[/tex] will be [tex]\( (-3, 2) \)[/tex].

4. Reflection across the line [tex]\( y = -x \)[/tex]:
- Reflecting across the line [tex]\( y = -x \)[/tex], both the [tex]\( x \)[/tex]-coordinate and [tex]\( y \)[/tex]-coordinate are swapped and their signs are changed.
- The coordinates of the vertex [tex]\( (2, -3) \)[/tex] reflected across the line [tex]\( y = -x \)[/tex] will be [tex]\( (3, -2) \)[/tex].

Given the vertex [tex]\( (2, -3) \)[/tex], the reflections produce the following image vertices:
- Across the [tex]\( x \)[/tex]-axis: [tex]\( (2, 3) \)[/tex]
- Across the [tex]\( y \)[/tex]-axis: [tex]\( (-2, -3) \)[/tex]
- Across the line [tex]\( y = x \)[/tex]: [tex]\( (-3, 2) \)[/tex]
- Across the line [tex]\( y = -x \)[/tex]: [tex]\( (3, -2) \)[/tex]

Thus, the answer to the question is as follows:

1. A reflection of [tex]\( \triangle R S T \)[/tex] across the [tex]\( x \)[/tex]-axis will result in the vertex [tex]\( (2, 3) \)[/tex].
2. A reflection of [tex]\( \triangle R S T \)[/tex] across the [tex]\( y \)[/tex]-axis will result in the vertex [tex]\( (-2, -3) \)[/tex].
3. A reflection of [tex]\( \triangle R S T \)[/tex] across the line [tex]\( y = x \)[/tex] will result in the vertex [tex]\( (-3, 2) \)[/tex].
4. A reflection of [tex]\( \triangle R S T \)[/tex] across the line [tex]\( y = -x \)[/tex] will result in the vertex [tex]\( (3, -2) \)[/tex].
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