Question
The rotation function mapping for a 90 degree clockwise rotation about the origin is equivalent to which other transformation mapping?
A reflection over the line y=x
A 270 degrees counterclockwise rotation
A dilation with a scale factor of 1/4
A translation
A reflection over the line y=x
A 270 degrees counterclockwise rotation
A dilation with a scale factor of 1/4
A translation
Answers
GPT-4o mini
A 90-degree clockwise rotation about the origin can be represented by the transformation \( (x, y) \mapsto (y, -x) \). To determine which of the given options corresponds to this transformation, we can analyze each one:
1. **A reflection over the line \( y = x \)**: This transformation maps \( (x, y) \) to \( (y, x) \), which is not the same as a 90-degree clockwise rotation.
2. **A 270 degrees counterclockwise rotation**: This transformation can be represented as \( (x, y) \mapsto (-y, x) \). To check equivalency, note that a 270-degree counterclockwise rotation is equivalent to a 90-degree clockwise rotation. Thus, this is the correct mapping.
3. **A dilation with a scale factor of 1/4**: This transformation scales a point \( (x, y) \) to \( \left(\frac{x}{4}, \frac{y}{4}\right) \), which does not involve rotation. Hence, it's not equivalent.
4. **A translation**: A translation shifts a point by adding constants to its coordinates, which does not represent a rotation.
Based on the analysis, the correct answer is:
**A 270 degrees counterclockwise rotation**.
1. **A reflection over the line \( y = x \)**: This transformation maps \( (x, y) \) to \( (y, x) \), which is not the same as a 90-degree clockwise rotation.
2. **A 270 degrees counterclockwise rotation**: This transformation can be represented as \( (x, y) \mapsto (-y, x) \). To check equivalency, note that a 270-degree counterclockwise rotation is equivalent to a 90-degree clockwise rotation. Thus, this is the correct mapping.
3. **A dilation with a scale factor of 1/4**: This transformation scales a point \( (x, y) \) to \( \left(\frac{x}{4}, \frac{y}{4}\right) \), which does not involve rotation. Hence, it's not equivalent.
4. **A translation**: A translation shifts a point by adding constants to its coordinates, which does not represent a rotation.
Based on the analysis, the correct answer is:
**A 270 degrees counterclockwise rotation**.