To determine the coordinates of point [tex]\( S \)[/tex] after a [tex]\( 270^\circ \)[/tex] rotation around the origin, it's helpful to know the properties of rotation transformations. A rotation of [tex]\( 270^\circ \)[/tex] (or equivalently, [tex]\(-90^\circ\)[/tex]) about the origin transforms any point [tex]\((x, y)\)[/tex] to [tex]\((y, -x)\)[/tex].
Here, the original coordinates of point [tex]\( S \)[/tex] are [tex]\((-2, -4)\)[/tex]. Applying the [tex]\( 270^\circ \)[/tex] rotation transformation to point [tex]\( S \)[/tex]:
1. Identify the original coordinates: [tex]\((x, y) = (-2, -4)\)[/tex].
2. Apply the transformation rule for a [tex]\( 270^\circ \)[/tex] rotation: [tex]\((x, y) \rightarrow (y, -x)\)[/tex].
Now substituting the values:
- [tex]\( y = -4 \)[/tex]
- [tex]\( x = -2 \)[/tex]
Using the transformation rule:
- The new [tex]\( x \)[/tex]-coordinate will be [tex]\( y = -4 \)[/tex].
- The new [tex]\( y \)[/tex]-coordinate will be [tex]\( -x = -(-2) = 2 \)[/tex].
Thus, the coordinates of [tex]\( S \)[/tex] after a [tex]\( 270^\circ \)[/tex] rotation are [tex]\((-4, 2)\)[/tex].
Based on this transformation, the coordinates of [tex]\( S \)[/tex] are:
[tex]\[ \boxed{(-4, 2)} \][/tex]
