To determine the new positions of vertices \( P \) and \( R \) after the rotation that moves \( Q \) to \( Q' \), we need to first establish the angle and direction of the rotation.
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Identifying the rotation: The original point \( Q(-4, -9) \) is rotated to \( Q'(-9, 4) \). We can find the vector that represents the rotation by observing the coordinates of these points.
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Establishing the rotation center: The center of rotation can be assumed to be at the origin (0, 0). Let's first confirm whether this rotation is indeed around the origin by calculating the distance from the origin to both points \( Q \) and \( Q' \) and checking their relative angles.
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Rotation analysis: The transformation of point \( Q \) to \( Q' \) can be expressed in terms of a rotation matrix \( R(\theta) \): \[ R(\theta) = \begin{pmatrix} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{pmatrix} \]
The specific coordinates post-rotation will mean we need to find the angle \( \theta \) that relates \( Q \) and \( Q' \).
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To analyze the points:
- \( Q \to Q' \) can be done either through \( 90^\circ \) counterclockwise or a specific angle found through the arctan function.
- To verify the transformation, we can visually verify how the points relate on the Cartesian plane and deduce the angle of rotation based on the positions.
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Calculating new positions: Given the rotation is likely \( 90^\circ \) counterclockwise, we can use the counterclockwise rotation transformation:
- For any point \( (x, y) \): \[ P' = (-y, x) \] Thus we perform the transformation for both \( P \) and \( R \):
- For \( P(-4, 0) \): \[ P' = (0, -4) \]
- For \( R(4, -4) \): \[ R' = (4, 4) \]
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Conclusion: Hence the images of the other two vertices are:
- \( P' = (0, -4) \)
- \( R' = (4, 4) \).
Now checking against the options given:
- The correct image is \( P′(0,−4) \) and \( R′(−4,4) \).
Therefore, the answer is: P′(0,−4) and R′(−4,4).
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