To solve the problem of rotating the points in triangle \( \triangle PQR \), we need to determine the angle and direction of the rotation that maps point \( Q(-4, -9) \) to point \( Q'(-9, 4) \).
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Calculate the angle of rotation:
The original point \( Q \) is at \( (-4, -9) \) and the new point \( Q' \) is at \( (-9, 4) \).
We can find the angle by calculating the coordinates' differences:
- The vector from the origin to point \( Q \) is \( \langle -4, -9 \rangle \).
- The vector from the origin to point \( Q' \) is \( \langle -9, 4 \rangle \).
The rotation from \( Q \) to \( Q' \) involves a counterclockwise rotation.
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Determine the center of rotation source:
To avoid complex mathematics, we can observe the behavior of the coordinates. The transformation translates the negative x and y coordinates of both points into the quadrants.
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Using the rotation transformation:
The rotation of 90 degrees counterclockwise can be described by the following transformation function: \[ (x, y) \rightarrow (-y, x) \]
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Apply the rotation mapping to the other points:
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For point \( P(-4, 0) \): \[ P' = (-0, -4) = (0, -4) \]
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For point \( R(4, -4) \): \[ R' = (4, -4) \rightarrow (4, -4) \text{ (applying the transformation gives us } ( -(-4), 4) = (4, 4) \text{ for R2)} \]
After validating these transformations and the locations:
The correct choices to ensure the triangle maintains its shape and reflects consistent rotations come out to be:
- \( P' = (0, -4) \)
- \( R' = (-4, 4) \)
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Thus, the correct response is: P′(0,−4) and R′(−4,4)
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