To determine the coordinates of [tex]\( S' \)[/tex] after the triangle is transformed according to the rule [tex]\( R_{0,270^\circ} \)[/tex], we need to apply a 270-degree counterclockwise rotation to the point [tex]\( S(-2, -4) \)[/tex].
The rule for rotating a point [tex]\( (x, y) \)[/tex] counterclockwise by 270 degrees around the origin is:
[tex]\[ (x, y) \rightarrow (y, -x) \][/tex]
Let’s apply this transformation to the coordinates of point [tex]\( S \)[/tex]:
1. Start with the original coordinates of [tex]\( S \)[/tex], which are [tex]\(-2\)[/tex] for [tex]\( x \)[/tex] and [tex]\(-4\)[/tex] for [tex]\( y \)[/tex].
2. According to the rule [tex]\( (x, y) \rightarrow (y, -x) \)[/tex]:
- The new [tex]\( x \)[/tex]-coordinate will be the original [tex]\( y \)[/tex]-coordinate: [tex]\( y = -4 \)[/tex].
- The new [tex]\( y \)[/tex]-coordinate will be the negative of the original [tex]\( x \)[/tex]-coordinate: [tex]\( -x = -(-2) = 2 \)[/tex].
Therefore, the new coordinates of [tex]\( S' \)[/tex] after the transformation are:
[tex]\[ S'(-4, 2) \][/tex]
So, the coordinates of [tex]\( S' \)[/tex] are [tex]\((-4, 2)\)[/tex].
Among the given options:
- [tex]\((-4, 2)\)[/tex]
- [tex]\((-2, 4)\)[/tex]
- [tex]\((2, 4)\)[/tex]
- [tex]\((4, -2)\)[/tex]
The correct answer is:
[tex]\((-4, 2)\)[/tex]
