In the coordinate plane, what is the rule for a $180°$ rotation about the origin that maps a point $(x,y)$ to its image?

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

Answer: (C)

Verified step by step guidance

1. Recall that a rotation of 180\(\degree\) about the origin in the coordinate plane means turning every point around the origin by 180\(\degree\) in the counterclockwise direction.
2. Use the rotation formulas for a point \((x, y)\) rotated by an angle \(\theta\) about the origin: \[\left(x', y'\right) = \left(x \cos \theta - y \sin \theta, x \sin \theta + y \cos \theta\right)\]
3. Substitute \(\theta = 180\degree\) into the formulas. Recall that \(\cos 180\degree = -1\) and \(\sin 180\degree = 0\), so the formulas become: \[x' = x \times (-1) - y \times 0 = -x\] \[y' = x \times 0 + y \times (-1) = -y\]
4. Therefore, the image of the point \((x, y)\) after a 180\(\degree\) rotation about the origin is \((-x, -y)\).
5. Compare this result with the given options to identify the correct mapping rule for the 180\(\degree\) rotation.