Question

a pentagon is transformed according to the rule (r_{0,180^{circ}}). which is another way to state the transformation?
((x,y)\to(-x, -y)
(x,y)\to(-y,-x)
(x,y)\to(x, -y)
(x,y)\to(-x,y))

Explanation:

Step1: Recall rotation rule

A rotation of $180^{\circ}$ about the origin $(0,0)$ in a coordinate - plane has the transformation rule.
For a point $(x,y)$ rotated $180^{\circ}$ about the origin, the new coordinates $(x',y')$ are given by $(x',y')=(-x,-y)$. The notation $R_{0,180^{\circ}}$ represents a rotation of $180^{\circ}$ about the origin. So the transformation rule for $R_{0,180^{\circ}}$ is $(x,y)\to(-x,-y)$.

Answer:

$(x,y)\to(-x,-y)$