To determine which transformation corresponds to the rule \((x, y) \rightarrow (x, y)\), we need to understand what this transformation represents.
1. Understanding the Transformation:
- The rule \((x, y) \rightarrow (x, y)\) means that each point \((x, y)\) in the original figure remains exactly at the same coordinates in the transformed figure.
- This implies that there is no change in position of any point in the figure.
2. Identifying the Types of Rotations:
- \(R_{0^{\circ}}\): A rotation of 0 degrees around the origin does not change the position of any point.
- \(R_{90^{\circ}}\): A rotation of 90 degrees around the origin would change the position of the points from \((x, y)\) to \((-y, x)\).
- \(R_{180^{\circ}}\): A rotation of 180 degrees around the origin would change the position of the points from \((x, y)\) to \((-x, -y)\).
- \(R_{270^{\circ}}\): A rotation of 270 degrees around the origin would change the position of the points from \((x, y)\) to \((y, -x)\).
- \(R_{360^{\circ}}\): A rotation of 360 degrees around the origin brings each point back to its original position \((x, y)\).
Since the rule \((x, y) \rightarrow (x, y)\) indicates that no change occurs in the position of the points, it corresponds to rotations of 0 degrees or 360 degrees, as both these rotations will place the points back in their original positions.
Hence, another way to state the transformation is:
[tex]\[ R_{0^{\circ}} \text{ or } R_{360^{\circ}} \][/tex]
Given the answer choices, the appropriate choice is:
[tex]\[ R_{0^{\circ}} \text{ or } R_{360^{\circ}} \][/tex]
With this logic, we can conclude that the correct option is:
[tex]\[ R_{0,360^{\circ}} \][/tex]
So the answer is:
[tex]\[ \boxed{R_{0,360^{\circ}}} \][/tex]
