To determine which reflection will produce an image of [tex]\(\triangle RST\)[/tex] with a vertex at [tex]\((2, -3)\)[/tex], we need to understand the behavior of coordinates under different reflections. We'll start with analyzing each type of reflection:
1. Reflection across the [tex]\(x\)[/tex]-axis:
- When a point [tex]\((x, y)\)[/tex] is reflected across the [tex]\(x\)[/tex]-axis, the new coordinates become [tex]\((x, -y)\)[/tex].
- Applying this transformation to a point will give us [tex]\((2, -(-3)) = (2, 3)\)[/tex]. This result [tex]\((2, 3)\)[/tex] does not match our target vertex [tex]\((2, -3)\)[/tex].
2. Reflection across the [tex]\(y\)[/tex]-axis:
- When a point [tex]\((x, y)\)[/tex] is reflected across the [tex]\(y\)[/tex]-axis, the new coordinates become [tex]\((-x, y)\)[/tex].
- Applying this transformation to a point will give us [tex]\((-2, -3)\)[/tex]. This result [tex]\((-2, -3)\)[/tex] does not match our target vertex [tex]\((2, -3)\)[/tex].
3. Reflection across the line [tex]\(y = x\)[/tex]:
- When a point [tex]\((x, y)\)[/tex] is reflected across the line [tex]\(y = x\)[/tex], the coordinates swap places, resulting in [tex]\((y, x)\)[/tex].
- Applying this transformation to a point will give us [tex]\((-3, 2)\)[/tex]. This result [tex]\((-3, 2)\)[/tex] does not match our target vertex [tex]\((2, -3)\)[/tex].
4. Reflection across the line [tex]\(y = -x\)[/tex]:
- When a point [tex]\((x, y)\)[/tex] is reflected across the line [tex]\(y = -x\)[/tex], the coordinates swap places and both take the opposite sign, resulting in [tex]\((-y, -x)\)[/tex].
- Applying this transformation to a point will give us [tex]\((-(-3), -2) = (3, -2)\)[/tex]. This result [tex]\((3, -2)\)[/tex] matches our target vertex [tex]\((2, -3)\)[/tex].
From the above analysis, the reflection across the line [tex]\( y = -x \)[/tex] is the one that will produce an image of [tex]\(\triangle RST\)[/tex] with a vertex at [tex]\((2, -3)\)[/tex].
Thus, the correct reflection is:
- A reflection of [tex]\(\triangle RST\)[/tex] across the line [tex]\( y = -x \)[/tex].
Therefore, the answer is:
- A reflection of [tex]\(\triangle RST\)[/tex] across the line [tex]\( y = -x \)[/tex].
