To determine which transformation is equivalent to the rule [tex]\((x, y) \rightarrow (x, y)\)[/tex], we need to analyze what this transformation means.
The rule [tex]\((x, y) \rightarrow (x, y)\)[/tex] indicates that each point [tex]\((x, y)\)[/tex] in the parallelogram maps to itself. In other words, the points are unchanged after the transformation. This implies that the parallelogram is essentially positioned exactly as it was initially.
To interpret this transformation from the perspective of rotating the figure, we consider the possible options:
1. [tex]\(R_{0,90^{\circ}}\)[/tex]: This represents a rotation of 90 degrees around the origin. A 90-degree rotation would move each point [tex]\((x, y)\)[/tex] to a new position [tex]\((-y, x)\)[/tex].
2. [tex]\(R_{0,180^{\circ}}\)[/tex]: This represents a rotation of 180 degrees around the origin. A 180-degree rotation would move each point [tex]\((x, y)\)[/tex] to a new position [tex]\((-x, -y)\)[/tex].
3. [tex]\(R_{0,270^{\circ}}\)[/tex]: This represents a rotation of 270 degrees around the origin. A 270-degree rotation would move each point [tex]\((x, y)\)[/tex] to a new position [tex]\((y, -x)\)[/tex].
4. [tex]\(R_{0,360^{\circ}}\)[/tex]: This represents a rotation of 360 degrees around the origin. A 360-degree rotation would move each point [tex]\((x, y)\)[/tex] back to its original position [tex]\((x, y)\)[/tex].
Since we determined that the transformation [tex]\((x, y) \rightarrow (x, y)\)[/tex] means all points remain in their original positions, the corresponding transformation is a 360-degree rotation.
Thus, another way to state the transformation [tex]\((x, y) \rightarrow (x, y)\)[/tex] is [tex]\(R_{0,360^{\circ}}\)[/tex].
