To solve this problem, we need to understand the transformation rule [tex]\( R_{0,180^\circ} \)[/tex]. This rule signifies that the pentagon is rotated by 180 degrees around the origin.
When any point [tex]\((x, y)\)[/tex] is rotated by 180 degrees about the origin, its coordinates change according to a specific pattern. Let's determine what that pattern is.
1. Initial point [tex]\((x, y)\)[/tex]: This is the original position of a point on the pentagon before any transformation.
2. Rotation by 180 degrees about the origin: In a 180-degree rotation:
- The x-coordinate of the point changes sign, meaning [tex]\(x\)[/tex] becomes [tex]\(-x\)[/tex].
- The y-coordinate of the point also changes sign, meaning [tex]\(y\)[/tex] becomes [tex]\(-y\)[/tex].
Therefore, the transformation of a point [tex]\((x, y)\)[/tex] after a 180-degree rotation around the origin results in the point [tex]\((-x, -y)\)[/tex].
So, the correct transformation rule is:
[tex]\[ (x, y) \rightarrow (-x, -y) \][/tex]
Thus, the correct answer to the question is:
[tex]\[ (x, y) \rightarrow (-x, -y) \][/tex]
