To find the correct transformation for a pentagon subjected to a rotation of [tex]\(180^\circ\)[/tex] around the origin, we consider the effect of this specific rotation on any point [tex]\((x, y)\)[/tex].
A rotation of [tex]\(180^\circ\)[/tex] around the origin in the coordinate plane transforms a point by reflecting it through the origin. Graphically, this switches the point across both the x-axis and the y-axis.
Let's analyze:
1. The original point [tex]\((x, y)\)[/tex] is rotated 180 degrees.
2. After [tex]\(180^\circ\)[/tex] rotation, the point [tex]\((x, y)\)[/tex] will be moved to a point which is directly opposite through the origin.
3. This results in both the x and y coordinates changing signs.
Thus, the transformed point [tex]\((x, y)\)[/tex] becomes [tex]\((-x, -y)\)[/tex].
Therefore, the correct transformation rule under a [tex]\(180^\circ\)[/tex] rotation around the origin is:
[tex]\[
(x, y) \rightarrow (-x, -y)
\][/tex]
So, the correct choice is:
[tex]\[
(x, y) \rightarrow (-x, -y)
\][/tex]
