To find the location of the point \( Q' \) after rotating the point \( Q(-5, -9) \) by \( -270^\circ \) about the origin, we can follow these steps:
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Understand the rotation direction: A rotation of \( -270^\circ \) is equivalent to a clockwise rotation of \( 270^\circ \) or a counterclockwise rotation of \( 90^\circ \).
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Use the rotation matrix: The rotation matrix for a counterclockwise rotation by an angle \( \theta \) is given by: \[ R(\theta) = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \ \sin(\theta) & \cos(\theta) \end{bmatrix} \] For \( \theta = 90^\circ \): \[ R(90^\circ) = \begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix} \]
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Apply the rotation matrix: We can apply this matrix to the coordinates of point \( Q(-5, -9) \): \[ Q' = R(90^\circ) \cdot \begin{bmatrix} -5 \ -9 \end{bmatrix} \] Calculating this gives: \[ Q' = \begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix} \cdot \begin{bmatrix} -5 \ -9 \end{bmatrix} = \begin{bmatrix} 0 \cdot (-5) + (-1) \cdot (-9) \ 1 \cdot (-5) + 0 \cdot (-9) \end{bmatrix} = \begin{bmatrix} 9 \ -5 \end{bmatrix} \]
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Locate \( Q' \): Therefore, the coordinates of the point \( Q' \) after the rotation are: \[ Q' = (9, -5) \]
Thus, the location of the point \( Q' \) after rotating \( Q(-5, -9) \) by \( -270^\circ \) about the origin is \( \boxed{(9, -5)} \).