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:
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Dilation with a scale factor of \( \frac{1}{4} \): This scales the distance from the origin but does not change the angle. Not equivalent.
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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.
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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.
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Translation: Translations move points without altering their orientation, distance, or angle. Thus, not equivalent.
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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.