To determine which transformation corresponds to a rotation of 180 degrees around the origin, let's carefully consider the geometrical effect of such a rotation:
1. Identify the Transformation:
- A rotation of 180 degrees around the origin (denoted [tex]\( R_{0,180^{\circ}} \)[/tex]) changes the position of each point on the coordinate plane by rotating them around the origin.
2. Original Point:
- Let’s start with a generic point [tex]\((x, y)\)[/tex] on the plane.
3. Expected Outcome After Transformation:
- When the point [tex]\((x, y)\)[/tex] is rotated 180 degrees around the origin, it will be directly opposite to its initial position, on the other side of the origin.
- This means that both the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] coordinates will change their signs.
4. Mathematical Representation:
- The transformation can be mathematically represented as [tex]\((x, y) \rightarrow (-x, -y)\)[/tex].
Now, let’s look at the given choices to find which matches our derived transformation:
1. [tex]\((x, y) \rightarrow (-x, -y)\)[/tex]
- This choice correctly represents the transformation rule where both the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] coordinates change their signs.
2. [tex]\((x, y) \rightarrow (-y, -x)\)[/tex]
- This choice incorrectly swaps the coordinates and changes their signs.
3. [tex]\((x, y) \rightarrow (x, -y)\)[/tex]
- This choice only changes the sign of the [tex]\(y\)[/tex]-coordinate and leaves the [tex]\(x\)[/tex]-coordinate unchanged.
4. [tex]\((x, y) \rightarrow (-x, y)\)[/tex]
- This choice only changes the sign of the [tex]\(x\)[/tex]-coordinate and leaves the [tex]\(y\)[/tex]-coordinate unchanged.
Considering the correct transformation for a 180-degree rotation around the origin, the best choice is:
[tex]\[
(x, y) \rightarrow (-x, -y)
\][/tex]
Therefore, the answer to the question is:
[tex]\[
(x, y) \rightarrow (-x, -y)
\][/tex]
