To determine the rule for rotating a point [tex]\((x, y)\)[/tex] by 180 degrees clockwise about the origin, we need to understand the effect of this rotation on the coordinates. Let’s go through the process step by step:
1. Understanding Rotation by 180 Degrees:
- Rotating a point by 180 degrees clockwise (or counterclockwise, since 180 degrees clockwise is equivalent to 180 degrees counterclockwise) involves a transformation that essentially flips the point to the opposite quadrant.
- This transformation changes both the x- and y-coordinates to their negatives.
2. Coordinate Transformation:
- Initially, the point is [tex]\((x, y)\)[/tex].
- After a 180-degree rotation, each coordinate is flipped to the opposite sign.
- Thus, [tex]\(x\)[/tex] becomes [tex]\(-x\)[/tex] and [tex]\(y\)[/tex] becomes [tex]\(-y\)[/tex].
3. Writing the Rule:
- The rule for rotating the point [tex]\((x, y)\)[/tex] by 180 degrees clockwise is given by:
[tex]\[
(x, y) \rightarrow (-x, -y)
\][/tex]
4. Identifying the Correct Option:
- We need to find which option matches our derived rule.
- Option D states: [tex]\((x, y) \rightarrow (-x, -y)\)[/tex].
Therefore, the correct rule for rotating the point [tex]\((x, y)\)[/tex] by 180 degrees clockwise about the origin is given by Option D.
So, the answer is:
[tex]\[
\boxed{D}
\][/tex]
