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A vertex of PQR is rotated from Q (-4, -9) to Q' (-9, 4). Use rotation function mapping to determine where the image of the other two vertices P (-4, 0) and R (4, -4), should be placed.

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To determine the new positions of the vertices \( P \) and \( R \) after the rotation of vertex \( Q \) to \( Q' \), we need to first find the center and angle of rotation used to map \( Q \) to \( Q' \).

  1. Identify the center of rotation: The center of rotation can commonly be identified as the midpoint of \( Q \) and \( Q' \). Thus, we calculate the midpoint \( M \) between \( Q(-4, -9) \) and \( Q'(-9, 4) \):

    \[ M_x = \frac{-4 + (-9)}{2} = \frac{-13}{2} = -6.5 \] \[ M_y = \frac{-9 + 4}{2} = \frac{-5}{2} = -2.5 \]

    Therefore, the center of rotation is at: \[ M(-6.5, -2.5) \]

  2. Calculate the angle of rotation: To find the angle of rotation, we can use the vector from \( M \) to \( Q \) and the vector from \( M \) to \( Q' \).

    • Vector from \( M \) to \( Q \): \[ V_{MQ} = Q - M = (-4, -9) - (-6.5, -2.5) = (2.5, -6.5) \]

    • Vector from \( M \) to \( Q' \): \[ V_{MQ'} = Q' - M = (-9, 4) - (-6.5, -2.5) = (-2.5, 6.5) \]

    Next, find the angle between these two vectors. The angle can be calculated using the dot product and magnitudes. However, looking at the vectors \( V_{MQ} \) and \( V_{MQ'} \), we observe that the vectors are orthogonal and their magnitudes are equal, indicating a 90-degree rotation.

  3. Rotate the points \( P \) and \( R \): We will apply a 90-degree rotation matrix around the center \( M(-6.5, -2.5) \).

    The rotation matrix for a 90-degree rotation counterclockwise is: \[ R = \begin{pmatrix} 0 & -1 \ 1 & 0 \end{pmatrix} \]

    We can express the rotation for a point \( (x, y) \) around a center \( (h, k) \) as follows:

    \[ \begin{pmatrix} x' \ y' \end{pmatrix}

    \begin{pmatrix} h + (y - k) \ k - (x - h) \end{pmatrix} \]

    • For point \( P(-4, 0) \):

      \[ P' = \begin{pmatrix} -6.5 + (0 + 2.5) \ -2.5 - (-4 + 6.5) \end{pmatrix} = \begin{pmatrix} -6.5 + 2.5 \ -2.5 - 2.5 \end{pmatrix} = \begin{pmatrix} -4 \ -5 \end{pmatrix} \]

    • For point \( R(4, -4) \):

      \[ R' = \begin{pmatrix} -6.5 + (-4 + 2.5) \ -2.5 - (4 + 6.5) \end{pmatrix} = \begin{pmatrix} -6.5 - 1.5 \ -2.5 - 10.5 \end{pmatrix} = \begin{pmatrix} -8 \ -13 \end{pmatrix} \]

  4. Final results: After applying the 90-degree rotation around the center \( (-6.5, -2.5) \): \[ P'(-4, -5) \quad \text{and} \quad R'(-8, -13) \]

So, the images of the vertices after rotation are:

  • \( Q' = (-9, 4) \)
  • \( P' = (-4, -5) \)
  • \( R' = (-8, -13) \)