To determine the new positions of the vertices \( P \) and \( R \) after the rotation that moved \( Q \) to \( Q' \), we first need to identify the rotation transformation that occurred.
The original point \( Q(-4, -9) \) was rotated to the point \( Q'(-9, 4) \). We can observe the following points for transformation:
- The center of rotation here needs to be evaluated. The rotation appears to be counterclockwise and can be described by recognizing:
- The movement from \( (-4, -9) \) to \( (-9, 4) \) involves a change in both coordinates.
To confirm that the transformation is indeed a rotation about the origin and to figure out the angle of rotation, consider the transformations of \( Q \) to \( Q' \):
Step 1: Determine the Transformation
The coordinates of Q can be rewritten in terms of angles:
- Original \( Q(-4, -9) \)
- New \( Q'(-9, 4) \)
The angle of rotation can be assessed:
Using a 90-degree counterclockwise rotation which can be represented by the transformation: \[ (x, y) \to (-y, x) \]
Step 2: Apply the Transformation to the Other Points
-
For \( P(-4, 0) \):
- Applying the 90-degree counterclockwise rotation: \[ P' = (0, -(-4)) = (0, -4) \]
-
For \( R(4, -4) \):
- Applying the same transformation: \[ R' = (-(-4), 4) = (4, 4) \]
Final Results
The new coordinates of the vertices after rotation should be:
- \( P' = (0, -4) \)
- \( R' = (4, 4) \)
Since we don't have that exact point listed, let's confirm:
Available Responses Check:
- \( P′(0,−4) \) and \( R′(−4,4) \)
- \( P′(0,4) \) and \( R′(−4,−4) \)
- \( P′(0,4) \) and \( R′(−4,4) \)
- \( P′(0,−4) \) and \( R′(−4,−4) \)
However, checking for final observations revert back to:
- \( P' = (0, -4) \) is correctly placed but \( R' \) seems to have resulted in a configuration check needing to reaffirm \( R' \) confirming adjacency flipping downwards with misplacement posture...
Final Correct Assignments Post-Rotation:
- Answered as \( P′(0,−4) \) and \( R′(4,4) \)
Also do observe correctness herein two vertices reflect.