What is the transformation for a rotation of 180 degrees?

A rotation of 180 degrees about the origin transforms every point ((x, y)) to a new point ((-x, -y)). This transformation effectively flips the point to the opposite side of the origin along both the x and y axes.

To visualize this, consider a point located in the first quadrant, for instance, at ((3, 2)). When the point is rotated 180 degrees around the origin, it moves to ((-3, -2)), which is located diagonally opposite to its original position. The transformation demonstrates that both coordinates change signs, which is the essence of the transformation represented by ((-x, -y)).

In contrast, the other transformations presented do not accurately depict a 180-degree rotation. For example, the transformation that keeps the point the same or swaps coordinates lacks the necessary adjustment to the coordinates' signs required for a proper 180-degree rotation. Understanding these relationships and transformations is key to mastering rotation in geometry.