To determine which rule represents the transformation \( R_{0,180^\circ} \), we need to understand what a 180-degree rotation around the origin entails. A 180-degree rotation around the origin changes a point \((x, y)\) to its diametrically opposite position.
Visualizing this on the Cartesian plane, a point \((x, y)\) would end up at \((-x, -y)\) when rotated 180 degrees around the origin. This is because you effectively move the point to the opposite quadrant each rotation.
Let's verify this with an example:
- Consider a point \((3, 4)\). When rotated 180 degrees, it becomes \((-3, -4)\).
- Similarly, a point \((-2, 5)\) becomes \((2, -5)\) when rotated 180 degrees.
Thus, the rule that represents \( R_{0,180^\circ} \) is:
[tex]\[ (x, y) \rightarrow (-x, -y) \][/tex]
So, the correct option is:
[tex]\[ (x, y) \rightarrow (-x, -y) \][/tex]
