To determine which rotations could have transformed the vertex from (3, –2) to (2, 3), we can apply the rotation transformations to the original vertex.
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Rotation \( R_{0, 90^\circ} \): \[ x' = -y, \quad y' = x \] Applying this to (3, –2): \[ x' = -(-2) = 2, \quad y' = 3 \] Resulting point: (2, 3) - This is a valid transformation.
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Rotation \( R_{0, 180^\circ} \): \[ x' = -x, \quad y' = -y \] Applying this to (3, –2): \[ x' = -3 = -3, \quad y' = -(-2) = 2 \] Resulting point: (-3, 2) - This is not a valid transformation.
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Rotation \( R_{0, 270^\circ} \): \[ x' = y, \quad y' = -x \] Applying this to (3, –2): \[ x' = -2, \quad y' = -3 \] Resulting point: (-2, -3) - This is not a valid transformation.
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Rotation \( R_{0, -90^\circ} \): \[ x' = y, \quad y' = -x \] Applying this to (3, –2): \[ x' = -2, \quad y' = -3 \] Resulting point: (-2, -3) - This is not a valid transformation.
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Rotation \( R_{0, -270^\circ} \) (equivalent to \( R_{0, 90^\circ} \)): \[ x' = -y, \quad y' = x \] Applying this to (3, –2): \[ x' = -(-2) = 2, \quad y' = 3 \] Resulting point: (2, 3) - This is a valid transformation.
Based on this analysis, the transformations that could have taken place are:
- \( R_{0, 90^\circ} \)
- \( R_{0, -270^\circ} \)
So, the correct options are:
- R0, 90°
- R0, –270°