To determine which transformation corresponds to the rule \((x, y) \rightarrow (x, y)\), we should consider what this transformation represents geometrically.
1. Understanding the Transformation:
- The rule \((x, y) \rightarrow (x, y)\) indicates that the coordinates of each point remain unchanged after the transformation.
- This means that after applying the transformation, each point in the parallelogram stays exactly where it originally was.
2. Interpreting the Identity Transformation:
- When a point \((x, y)\) does not change its position, this implies that no actual transformation has occurred, or if a transformation has occurred, it effectively leaves every point in its original position.
- The only rotation that leaves every point in its original position occurs when the rotation completes a full circle, which amounts to \(360^\circ\).
3. Rotation Notation:
- We commonly denote rotations around the origin with \(R_{0, \theta}\), where \(\theta\) is the angle of rotation.
- Therefore, a rotation that leaves the points unchanged would be a \(360^\circ\) rotation, written as \(R_{0, 360^\circ}\).
4. Comparing Options:
- \(R_{0, 90^\circ}\): This rotates points by 90 degrees counterclockwise, changing their positions.
- \(R_{0, 180^\circ}\): This rotates points by 180 degrees, also changing their positions.
- \(R_{0, 270^\circ}\): This rotates points by 270 degrees counterclockwise, changing their positions.
- \(R_{0, 360^\circ}\): This rotates points by 360 degrees, returning them to their original positions.
Given the options, the correct transformation that describes the rule \((x, y) \rightarrow (x, y)\) is indeed the rotation by \(360\) degrees.
Thus, the correct answer is:
[tex]\[ R_{0, 360^\circ} \][/tex]
