Asked by Ironmarshy14
The rotation function mapping for a 90-degree clockwise rotation about the origin is equivalent to which other transformation mapping?(1 point) Responses a dilation with a scale factor of 14 a dilation with a scale factor of Start Fraction 1 over 2 End Fraction a 270 degrees counterclockwise rotation a 270 degrees counterclockwise rotation a translation a translation a reflection over the line y=x
Answers
Answered by
GPT-4o mini
A 90-degree clockwise rotation about the origin can be represented by the transformation mapping \((x, y) \to (y, -x)\).
To see how this rotation is equivalent to other transformations, we can compare it to the provided options:
1. **Dilation with a scale factor of \( \frac{1}{4} \)**: This scales the distance from the origin but does not change the angle. Not equivalent.
2. **Dilation with a scale factor of \( \frac{1}{2} \)**: Similar to option 1, this just scales distances and does not change the angle. Not equivalent.
3. **270 degrees counterclockwise rotation**: A 270-degree counterclockwise rotation about the origin corresponds to the transformation mapping \((x, y) \to (-y, x)\), which is the same as the 90-degree clockwise rotation, as they result in the same final positions for the points. **This is equivalent.**
4. **Translation**: Translations move points without altering their orientation, distance, or angle. Thus, not equivalent.
5. **Reflection over the line \(y=x\)**: This transforms points as follows: \((x, y) \to (y, x)\). It does not result in a 90-degree clockwise rotation and thus is not equivalent.
Among the options given, the correct response that is equivalent to a 90-degree clockwise rotation about the origin is:
**A 270 degrees counterclockwise rotation.**
To see how this rotation is equivalent to other transformations, we can compare it to the provided options:
1. **Dilation with a scale factor of \( \frac{1}{4} \)**: This scales the distance from the origin but does not change the angle. Not equivalent.
2. **Dilation with a scale factor of \( \frac{1}{2} \)**: Similar to option 1, this just scales distances and does not change the angle. Not equivalent.
3. **270 degrees counterclockwise rotation**: A 270-degree counterclockwise rotation about the origin corresponds to the transformation mapping \((x, y) \to (-y, x)\), which is the same as the 90-degree clockwise rotation, as they result in the same final positions for the points. **This is equivalent.**
4. **Translation**: Translations move points without altering their orientation, distance, or angle. Thus, not equivalent.
5. **Reflection over the line \(y=x\)**: This transforms points as follows: \((x, y) \to (y, x)\). It does not result in a 90-degree clockwise rotation and thus is not equivalent.
Among the options given, the correct response that is equivalent to a 90-degree clockwise rotation about the origin is:
**A 270 degrees counterclockwise rotation.**
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