To answer this question, we need to understand what the rule [tex]\( R_{0,180^\circ} \)[/tex] entails. The notation [tex]\( R_{0,180^\circ} \)[/tex] typically represents a rotation of 180 degrees around the origin (0, 0).
### Step-by-Step Explanation:
1. Understanding Rotation by 180 Degrees:
- When an object is rotated 180 degrees around the origin in a Cartesian coordinate system, each point [tex]\((x, y)\)[/tex] on the object will be moved to a new position.
- Specifically, under a 180-degree rotation, the coordinates [tex]\((x, y)\)[/tex] are transformed to [tex]\((-x, -y)\)[/tex]. This happens because rotating 180 degrees flips the signs of both the x-coordinate and the y-coordinate.
2. Verification of the Rule:
- The transformation from [tex]\((x, y)\)[/tex] to [tex]\((-x, -y)\)[/tex] effectively maps each point to its opposite point across the origin.
- For example:
- A point [tex]\((3, 4)\)[/tex] would be transformed to [tex]\((-3, -4)\)[/tex].
- A point [tex]\((-2, -5)\)[/tex] would transform to [tex]\((2, 5)\)[/tex].
3. Comparing Options:
- The first option, [tex]\((x, y) \rightarrow (-x, -y)\)[/tex], matches exactly with the rule we identified for a 180-degree rotation.
- The second option, [tex]\((x, y) \rightarrow (-y - x)\)[/tex], does not make sense within this context. Both coordinates being negative does not relate to 180-degree rotation.
- The third option, [tex]\((x, y) \rightarrow (x, -y)\)[/tex], describes reflection over the x-axis, not a rotation.
- The fourth option, [tex]\((x, y) \rightarrow (-x, y)\)[/tex], describes reflection over the y-axis, not a rotation.
4. Conclusion:
- The correct transformation that represents a 180-degree rotation around the origin is [tex]\((x, y) \rightarrow (-x, -y)\)[/tex].
Therefore, the answer is:
[tex]\[ (x, y) \rightarrow (-x, -y) \][/tex]
