To determine the nature of the transformation given by the rule [tex]\((x, y) \rightarrow (x, y)\)[/tex], let's analyze what each proposed transformation [tex]\( R_{0,\theta^{\circ}} \)[/tex] means:
1. [tex]\( R_{0,90^{\circ}} \)[/tex]: This represents a rotation of 90 degrees about the origin. The transformation would change any point [tex]\((x, y)\)[/tex] to the point [tex]\((-y, x)\)[/tex].
2. [tex]\( R_{0,180^{\circ}} \)[/tex]: This represents a rotation of 180 degrees about the origin. The transformation would change any point [tex]\((x, y)\)[/tex] to the point [tex]\((-x, -y)\)[/tex].
3. [tex]\( R_{0,270^{\circ}} \)[/tex]: This represents a rotation of 270 degrees about the origin. The transformation would change any point [tex]\((x, y)\)[/tex] to the point [tex]\((y, -x)\)[/tex].
4. [tex]\( R_{0,360^{\circ}} \)[/tex]: This represents a rotation of 360 degrees about the origin. Since 360 degrees is a full rotation, any point [tex]\((x, y)\)[/tex] would map back to itself, resulting in [tex]\((x, y) \rightarrow (x, y)\)[/tex].
Given the transformation rule [tex]\((x, y) \rightarrow (x, y)\)[/tex], the coordinates of any point remain unchanged. This is exactly what happens in a 360-degree rotation. Therefore, another way to state the transformation is:
[tex]\[ R_{0,360^{\circ}} \][/tex]
So the correct answer is:
[tex]\[ R_{0,360^{\circ}} \][/tex]
And the corresponding choice is:
[tex]\[ \boxed{R_{0,360^{\circ}}} \][/tex]
