Question
What is the image of the point \( (3,-6) \) after a rotation of \( 180^{\circ} \) counterclockwise about the origin?
Ask by Huang Pena. in the United States
Mar 10,2025
Upstudy AI Solution
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Answer
The image of the point \((3, -6)\) after a \(180^\circ\) rotation is \((-3, 6)\).
Solution
1. A \(180^\circ\) rotation about the origin transforms any point \((x, y)\) to \((-x, -y)\).
2. Given the point \((3, -6)\), we apply the transformation:
\[
(3, -6) \to (-3, 6)
\]
3. Therefore, the image of the point \((3, -6)\) after a \(180^\circ\) rotation is \((-3, 6)\).
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Simplify this solution The Deep Dive
To find the image of the point \( (3, -6) \) after a rotation of \( 180^{\circ} \) counterclockwise about the origin, we can apply the rotation transformation. A \( 180^{\circ} \) rotation about the origin effectively negates both the x-coordinate and y-coordinate of the point. Therefore, we calculate: \[ (3, -6) \rightarrow (-3, 6) \] Thus, the image of the point \( (3, -6) \) after the rotation is \( \boxed{(-3, 6)} \).
