To solve this problem, we need to determine the rule for rotating a point [tex]\( (x, y) \)[/tex] 180 degrees clockwise about the origin.
1. Understanding the Rotation:
- When you rotate a point [tex]\( (x, y) \)[/tex] 180 degrees clockwise around the origin, the new position of the point should be directly opposite to its original position relative to the origin.
2. Geometric Insight:
- Consider the point [tex]\( (x, y) \)[/tex]. Rotating it halfway around a full circle (180 degrees) will result in a point that is on the exact opposite side of the origin.
- This transformation means that both the x-coordinate and the y-coordinate will be inverted (sign changed).
3. Mathematical Representation:
- The transformation can be mathematically represented as [tex]\( (x, y) \)[/tex] mapping to [tex]\( (-x, -y) \)[/tex].
- This is because you need to go to the opposite side of the origin, taking both coordinates' negative values.
Based on this understanding, the correct choice from the given options is:
A. [tex]\( (x, y) \rightarrow (-x, -y) \)[/tex]
Therefore, the rule for rotating the point [tex]\((x, y)\)[/tex], 180 degrees clockwise about the origin, is [tex]\( (x, y) \rightarrow (-x, -y) \)[/tex]. The correct answer is option A.
