To determine which rule represents a 180-degree rotation around the origin, denoted [tex]\( R_{0,180^\circ} \)[/tex], we need to understand how a point [tex]\((x, y)\)[/tex] is transformed during this rotation.
Here's the step-by-step explanation:
1. Understanding 180-degree Rotation:
- A 180-degree rotation around the origin means that every point on the plane is rotated halfway around a circle centered at the origin.
2. Effect of 180-degree Rotation:
- Intuitively, when a point [tex]\((x, y)\)[/tex] is rotated 180 degrees around the origin, it ends up directly opposite to where it started. This means that the new point will have the same distance from the origin but in the opposite direction.
3. Coordinate Transformation:
- When we rotate a point [tex]\((x, y)\)[/tex] by 180 degrees, the new coordinates become [tex]\((-x, -y)\)[/tex]. This is because both the x and y coordinates flip to their opposites, effectively placing the point in the opposite quadrant of the coordinate plane.
4. Identifying the Correct Rule:
- Given the options:
1. [tex]\((x, y) \rightarrow (-y, x)\)[/tex]
2. [tex]\((x, y) \rightarrow (-x, -y)\)[/tex]
3. [tex]\((x, y) \rightarrow (y, -x)\)[/tex]
4. [tex]\((x, y) \rightarrow (y, x)\)[/tex]
- The only correct transformation for a 180-degree rotation, as explained, is [tex]\((x, y) \rightarrow (-x, -y)\)[/tex].
Therefore, the rule that represents [tex]\( R_{0,180^\circ} \)[/tex] is:
[tex]\[ (x, y) \rightarrow (-x, -y) \][/tex]
So the correct answer is:
[tex]\[ 2 \][/tex]
