Which of the following is the correct mapping rule for a $180°$ rotation about the origin in the coordinate plane?

• A. $(x,y)\to (-x,-y)$
• B. $(x,y)\to (y,-x)$
• C. $(x,y)\to (x,-y)$
• D. $(x,y)\to (-y,x)$

Answer: (A)

Verified step by step guidance

1. Recall that a rotation of 180 degrees about the origin in the coordinate plane means turning every point around the origin by half a full circle.
2. Understand that rotating a point \((x, y)\) by 180 degrees will place it directly opposite on the coordinate plane, which geometrically corresponds to negating both the \(x\) and \(y\) coordinates.
3. Express this transformation as a mapping rule: the point \((x, y)\) is mapped to \((-x, -y)\) after a 180 degree rotation about the origin.
4. Compare this mapping rule to the given options to identify which one matches the transformation \((x, y) \to (-x, -y)\).
5. Confirm that the correct mapping rule for a 180 degree rotation about the origin is \((x, y) \to (-x, -y)\), as it correctly represents the rotation.