Question

12 multiple choice 1 point what is the rule for a 180 degree clockwise/counterclockwise rotation? (x,y)→(−x,−y) (x,y)→(−y,−x) (x,y)→(y,−x) (x,y)→(−y,x) clear my selection

Explanation:

Step1: Recall rotation rules

For a \(180^{\circ}\) rotation (clockwise or counter - clockwise) about the origin, we can use the property of coordinate transformation.
Let's consider a point \((x,y)\) in the coordinate plane.

Step2: Analyze the transformation

If we rotate a point \((x,y)\) \(180^{\circ}\) about the origin, the new \(x\) - coordinate is the opposite of the original \(x\) - coordinate and the new \(y\) - coordinate is the opposite of the original \(y\) - coordinate.
Mathematically, if we have a point \(P(x,y)\) and we rotate it \(180^{\circ}\) about the origin \(O(0,0)\), using the rotation matrix \(

$$\begin{bmatrix}-1&0\\0& - 1\end{bmatrix}$$

\times

$$\begin{bmatrix}x\\y\end{bmatrix}$$

=

$$\begin{bmatrix}-x\\-y\end{bmatrix}$$

\)

Answer:

\((x,y)\to(-x, - y)\)