To determine which transformation is represented by the rule [tex]\((x, y) \rightarrow (y, -x)\)[/tex], we need to analyze how this rule affects a point on the coordinate plane.
1. Original Position: Start with a point [tex]\((x, y)\)[/tex].
2. Transformation: Apply the given rule [tex]\((x, y) \rightarrow (y, -x)\)[/tex].
- The x-coordinate of the original point [tex]\((x, y)\)[/tex] becomes the y-coordinate in the transformed point.
- The y-coordinate of the original point [tex]\((x, y)\)[/tex] becomes the negative of the x-coordinate in the transformed point.
Let's investigate what this transformation represents geometrically:
1. 90 Degrees Counterclockwise Rotation:
- When a point [tex]\((x, y)\)[/tex] is rotated 90 degrees counterclockwise around the origin, the coordinates of the point [tex]\((x, y)\)[/tex] are transformed to [tex]\((y, -x)\)[/tex].
- This matches the given transformation rule.
2. Checking Other Options:
- 180 Degrees Rotation: This would transform [tex]\((x, y)\)[/tex] to [tex]\((-x, -y)\)[/tex].
- 270 Degrees Rotation: This would transform [tex]\((x, y)\)[/tex] to [tex]\((-y, x)\)[/tex].
- 360 Degrees Rotation: This would keep the point [tex]\((x, y)\)[/tex] unchanged.
Given this analysis, the rule [tex]\((x, y) \rightarrow (y, -x)\)[/tex] corresponds to a 90-degree counterclockwise rotation around the origin.
Thus, the transformed quadrilateral ABCD according to the rule [tex]\((x, y) \rightarrow (y, -x)\)[/tex] is equivalent to the transformation [tex]\( R_{0, 90^{\circ}} \)[/tex].
The correct answer is:
[tex]\[ R_{0, 90^{\circ}}. \][/tex]
