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Which reflection will produce an image of [tex]\Delta RST[/tex] with a vertex at \((2, -3)\)?

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



Answer :

GinnyAnswer
To find the image of a vertex [tex]$\Delta R S T$[/tex] with coordinates [tex]$(2, -3)$[/tex] reflected across different lines and axes, let's go through each reflection step-by-step to determine the resulting coordinates.

1. Reflection across the x-axis:
- When reflecting a point across the x-axis, the x-coordinate remains the same, and the y-coordinate is negated.
- Starting with the point [tex]$(2, -3)$[/tex]:
- The x-coordinate is [tex]$2$[/tex].
- The y-coordinate is negated from [tex]$-3$[/tex] to [tex]$3$[/tex].
- Therefore, the reflection of [tex]$(2, -3)$[/tex] across the x-axis is [tex]$(2, 3)$[/tex].

2. Reflection across the y-axis:
- When reflecting a point across the y-axis, the y-coordinate remains the same, and the x-coordinate is negated.
- Starting with the point [tex]$(2, -3)$[/tex]:
- The x-coordinate is negated from [tex]$2$[/tex] to [tex]$-2$[/tex].
- The y-coordinate is [tex]$-3$[/tex].
- Therefore, the reflection of [tex]$(2, -3)$[/tex] across the y-axis is [tex]$(-2, -3)$[/tex].

3. Reflection across the line [tex]$y=x$[/tex]:
- When reflecting a point across the line [tex]$y=x$[/tex], the coordinates are swapped.
- Starting with the point [tex]$(2, -3)$[/tex]:
- Swap the coordinates to get [tex]$(-3, 2)$[/tex].
- Therefore, the reflection of [tex]$(2, -3)$[/tex] across the line [tex]$y=x$[/tex] is [tex]$(-3, 2)$[/tex].

4. Reflection across the line [tex]$y=-x$[/tex]:
- When reflecting a point across the line [tex]$y=-x$[/tex], the coordinates are swapped and negated.
- Starting with the point [tex]$(2, -3)$[/tex]:
- Swap the coordinates to get [tex]$(-3, 2)$[/tex], and then negate both coordinates.
- The x-coordinate becomes [tex]$3$[/tex] and the y-coordinate becomes [tex]$-2$[/tex].
- Therefore, the reflection of [tex]$(2, -3)$[/tex] across the line [tex]$y=-x$[/tex] is [tex]$(3, -2)$[/tex].

In conclusion:

- The reflection of [tex]$\Delta R S T$[/tex] with vertex [tex]$(2,-3)$[/tex] across the x-axis results in [tex]$(2, 3)$[/tex].
- The reflection across the y-axis results in [tex]$(-2, -3)$[/tex].
- The reflection across the line [tex]$y=x$[/tex] results in [tex]$(-3, 2)$[/tex].
- The reflection across the line [tex]$y=-x$[/tex] results in [tex]$(3, -2)$[/tex].

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