To determine the correct transformation of a pentagon according to the rule [tex]\( R_{0,180^{\circ}} \)[/tex], we need to understand what this rule represents.
The rule [tex]\( R_{0,180^{\circ}} \)[/tex] means we are rotating the point around the origin by 180 degrees. When you rotate a point [tex]\((x, y)\)[/tex] by 180 degrees around the origin, both the x-coordinate and the y-coordinate change signs. Specifically, the point [tex]\((x, y)\)[/tex] will move to [tex]\((-x, -y)\)[/tex].
Given the options:
1. [tex]\((x, y) \rightarrow (-x, -y)\)[/tex]
2. [tex]\((x, y) \rightarrow (-y, -x)\)[/tex]
3. [tex]\((x, y) \rightarrow (x, -y)\)[/tex]
4. [tex]\((x, y) \rightarrow (-x, y)\)[/tex]
By definition of the 180-degree rotation about the origin:
- Option 1: [tex]\((x, y) \rightarrow (-x, -y)\)[/tex] depicts the transformation accurately. Both coordinates change their signs, which is what happens during a 180-degree rotation around the origin.
- Option 2: [tex]\((x, y) \rightarrow (-y, -x)\)[/tex] does not correctly represent a 180-degree rotation, as the coordinates are swapped and their signs are changed.
- Option 3: [tex]\((x, y) \rightarrow (x, -y)\)[/tex] only changes the sign of the y-coordinate, which represents a reflection over the x-axis rather than a rotation.
- Option 4: [tex]\((x, y) \rightarrow (-x, y)\)[/tex] only changes the sign of the x-coordinate, which represents a reflection over the y-axis rather than a rotation.
Thus, the correct transformation according to [tex]\( R_{0,180^{\circ}} \)[/tex] is:
[tex]\[
(x, y) \rightarrow (-x, -y)
\][/tex]
Therefore, the answer is:
[tex]\((x, y) \rightarrow (-x, -y)\)[/tex]
