Let's break down the transformation rule [tex]\((x, y) \rightarrow (x, y)\)[/tex].
1. Understanding the Transformation:
- The given rule states that the coordinates of any point [tex]\((x, y)\)[/tex] remain exactly the same. This means that there is no change in the position of the point after the transformation.
2. Interpreting the Transformation in Terms of Rotation:
- Simple scenarios where coordinates [tex]\((x, y)\)[/tex] remain unchanged typically involve rotations that return the system to its original configuration.
- In the context of rotations about the origin, there are multiple standard rotations: [tex]\(90^\circ\)[/tex], [tex]\(180^\circ\)[/tex], [tex]\(270^\circ\)[/tex], and [tex]\(360^\circ\)[/tex].
3. Analyzing the Possible Rotations:
- [tex]\(R_{0,90^\circ}\)[/tex] typically results in [tex]\((x, y) \rightarrow (-y, x)\)[/tex], which changes the coordinates.
- [tex]\(R_{0,180^\circ}\)[/tex] usually results in [tex]\((x, y) \rightarrow (-x, -y)\)[/tex], changing the coordinates.
- [tex]\(R_{0,270^\circ}\)[/tex] rotates the point such that [tex]\((x, y) \rightarrow (y, -x)\)[/tex], again altering the coordinates.
- [tex]\(R_{0,360^\circ}\)[/tex] essentially returns the point to its original position because [tex]\(360^\circ\)[/tex] is a full rotation, which means [tex]\((x, y) \rightarrow (x, y)\)[/tex].
4. Conclusion:
- Since the transformation [tex]\((x, y) \rightarrow (x, y)\)[/tex] indicates that the coordinates do not change, it corresponds to a [tex]\(360^\circ\)[/tex] rotation.
Based on the above analysis, the correct answer is:
[tex]\[ R_{0,360^*} \][/tex]
Therefore, the correct answer is the option:
[tex]\[ \boxed{4} \][/tex]
