Rotating a point [tex]\( 90^\circ \)[/tex] about the origin transforms the point to a new location in the coordinate plane. To determine the exact transformation rule, let's analyze the given options step-by-step.
1. Rotation about the origin [tex]\(90^\circ\)[/tex] counterclockwise:
- Initially, if we have a point [tex]\( (x, y) \)[/tex], after rotation it will be moved to a new position. Specifically, for a [tex]\( 90^\circ \)[/tex] counterclockwise rotation, the point [tex]\( (x, y) \)[/tex] will transform according to a well-known geometric rule.
2. Coordinate Changes:
- The result of a [tex]\( 90^\circ \)[/tex] counterclockwise rotation around the origin changes a point [tex]\( (x, y) \)[/tex] to [tex]\( (-y, x) \)[/tex]. This can be visualized on the coordinate plane, where:
- The original x-coordinate becomes the new negative y-coordinate.
- The original y-coordinate becomes the new x-coordinate.
Given this understanding, let's match the correct transformation rule from the options provided:
- [tex]\( (x, y) \rightarrow (-x, -y) \)[/tex]: This is a reflection through the origin, not a [tex]\( 90^\circ \)[/tex] rotation.
- [tex]\( (x, y) \rightarrow (-y, x) \)[/tex]: This correctly represents the transformation resulting from a [tex]\( 90^\circ \)[/tex] counterclockwise rotation.
- [tex]\( (x, y) \rightarrow (-y, -x) \)[/tex]: This transformation does not align with the characteristics of a [tex]\( 90^\circ \)[/tex] counterclockwise rotation.
- [tex]\( (x, y) \rightarrow (y, -x) \)[/tex]: This transformation represents a [tex]\( 90^\circ \)[/tex] clockwise rotation, not counterclockwise.
Hence, the correct transformation rule that describes the [tex]\( 90^\circ \)[/tex] counterclockwise rotation about the origin is:
[tex]\((x, y) \rightarrow (-y, x)\)[/tex].
Therefore, the correct choice is:
[tex]\[
(x, y) \rightarrow (-y, x)
\][/tex]
