Question
Question What is the image of the point \( (-9,7) \) after a rotation of \( 180^{\circ} \) counterclockwise about the origin? Answer Attempt 1 out of 2
Ask by Hammond Cross. in the United States
Mar 10,2025
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Answer
The image of the point \((-9, 7)\) after a \(180^{\circ}\) counterclockwise rotation about the origin is \((9, -7)\).
Solution
1. A rotation of \(180^{\circ}\) about the origin transforms a point \((x, y)\) to \((-x, -y)\).
2. For the point \((-9, 7)\), applying the transformation we get:
\[
(-(-9), -(7)) = (9, -7)
\]
3. Thus, the image of \((-9, 7)\) after a \(180^{\circ}\) counterclockwise rotation about the origin is \((9, -7)\).
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Simplify this solution The Deep Dive
When you rotate a point \( (x, y) \) by \( 180^{\circ} \) counterclockwise about the origin, you can simply change the signs of both coordinates. For the point \( (-9, 7) \), the image after rotation becomes \( (9, -7) \). Just think of it as flipping the point to the opposite quadrant! A little trick for remembering this transformation is to focus on the idea of "180 degrees: all switches!" Both coordinates get flipped to their opposites. If you're ever unsure about the rotation, a handy visualization is to imagine the point traveling in a semicircle to land directly across from its original spot on the coordinate plane.
