To find the images of the vertices P and R after the rotation that maps Q to Q', we need to determine the angle of rotation and the center of rotation.
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Finding the angle of rotation: The initial point Q is at (-4, -9) and is rotated to Q' at (-9, 4). We can find the angle of rotation by calculating the vector from the center of rotation to these points.
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Calculating the distance: The distance from the center of rotation to Q and Q' must remain the same since rotations preserve distances.
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Rotation angle: We can visualize the transformation. The rotation can be determined by examining the difference in angles. The initial position gives the angle we can calculate using: \[ \theta_Q = \tan^{-1}\left(\frac{-9 - 0}{-4 - 0}\right) = \tan^{-1}\left(\frac{-9}{-4}\right) \] and for Q': \[ \theta_{Q'} = \tan^{-1}\left(\frac{4 - 0}{-9 - 0}\right) = \tan^{-1}\left(\frac{4}{-9}\right) \]
From these points, we can see the rotation around some origin will involve shifting the angles.
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Determine the new points P' and R': Since Q has been rotated and needs to maintain relative positioning:
For a rotation from left to top quadrant, often angles are rotated by 90° or similarly structured.
Consider the common transformation used in 90-degree rotations counterclockwise about the origin (which fits a (-4,-9) to (-9, 4) mapping): \[ \text{Rotating point}(x, y) \rightarrow (-y, x) \]
Applying this to the other two points:
- For P(-4, 0): \[ P' = (0, -4) \]
- For R(4, -4): \[ R' = (4, 4) \]
However, because you notice that R movement is still relative with regards to our rotation point moving leads one to an adjustment with the signs through calculating the geometry or using the known transformations produces here.
After carrying out the transformations, we identify:
- The correct answer is: P' (0, 4) and R' (-4, 4)
Thus confirming P moves after rotation up while R deflects downwards maintaining a rhombus shape post rotation in the quadrant processed.