To determine the correct transformation rule for [tex]\(R_{0,180^{\circ}}\)[/tex], we need to understand what it means to rotate a point around the origin by 180 degrees.
Here are the steps to solve this question:
1. Understand the Transformation:
- Rotating a point [tex]\((x, y)\)[/tex] by 180 degrees around the origin means turning the point halfway around the circle centered at the origin. Essentially, the point ends up on the opposite side of the origin, inverting both the x and y coordinates.
2. Determine the Coordinates after Rotation:
- If you rotate [tex]\((x, y)\)[/tex] by 180 degrees, the new coordinates will be [tex]\((-x, -y)\)[/tex]. This is because each coordinate essentially flips to the negative side of the axis.
3. Match with Given Options:
- We need to match our result [tex]\((-x, -y)\)[/tex] with one of the given transformations:
- [tex]\((x, y) \rightarrow (-x, -y)\)[/tex]
- [tex]\((x, y) \rightarrow (-y, -x)\)[/tex]
- [tex]\((x, y) \rightarrow (x, -y)\)[/tex]
- [tex]\((x, y) \rightarrow (-x, y)\)[/tex]
4. Identify the Correct Choice:
- It's evident from the steps above that [tex]\((-x, -y)\)[/tex] is the correct transformation rule for [tex]\(R_{0,180^{\circ}}\)[/tex], because it directly translates to flipping both coordinates.
Therefore, the correct transformation is:
[tex]\[
(x, y) \rightarrow (-x, -y)
\][/tex]
