9. Find the coordinates of each image point: a) \( D_{0,-2}:(1,3) \rightarrow(-2,-6) \) b) \( R_{o, 90^{\circ}}:(2,-4) \rightarrow(4,2) \) c) \( D_{(1,2), \frac{1}{2}}:(5,4) \rightarrow \) d) \( R_{(-2,-3),-90^{\circ}}:(2,-5) \rightarrow \)

Let's solve each part of Problem 9 step by step. ### **Part c: \( D_{(1,2), \frac{1}{2}}:(5,4) \rightarrow \)** **Transformation Description:** - **Dilation (Scaling)** centered at point \( (1,2) \) with a scale factor of \( \frac{1}{2} \). **Steps to Find the Image Point:** 1. **Identify the center of dilation:** \( C = (1,2) \). 2. **Original point:** \( P = (5,4) \). 3. **Scale factor:** \( k = \frac{1}{2} \). 4. **Apply the dilation formula:** \[ P' = C + k \cdot (P - C) \] Breaking it down: \[ x' = 1 + \frac{1}{2} \cdot (5 - 1) = 1 + \frac{1}{2} \cdot 4 = 1 + 2 = 3 \] \[ y' = 2 + \frac{1}{2} \cdot (4 - 2) = 2 + \frac{1}{2} \cdot 2 = 2 + 1 = 3 \] **Image Point:** \[ (3, 3) \] --- ### **Part d: \( R_{(-2,-3),-90^{\circ}}:(2,-5) \rightarrow \) \** **Transformation Description:** - **Rotation** centered at point \( (-2,-3) \) by \( -90^\circ \) (which is a clockwise rotation of 90 degrees). **Steps to Find the Image Point:** 1. **Identify the center of rotation:** \( C = (-2,-3) \). 2. **Original point:** \( P = (2,-5) \). 3. **Angle of rotation:** \( \theta = -90^\circ \). 4. **Translate the original point to the origin:** \[ P_{\text{translated}} = (2 - (-2), -5 - (-3)) = (4, -2) \] 5. **Apply the rotation matrix for \( -90^\circ \):** \[ \begin{pmatrix} \cos(-90^\circ) & -\sin(-90^\circ) \\ \sin(-90^\circ) & \cos(-90^\circ) \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \] Applying to \( (4, -2) \): \[ x' = 0 \cdot 4 + 1 \cdot (-2) = -2 \] \[ y' = -1 \cdot 4 + 0 \cdot (-2) = -4 \] 6. **Translate back to the original center:** \[ P' = (-2 + (-2), -3 + (-4)) = (-4, -7) \] **Image Point:** \[ (-4, -7) \] --- ### **Summary of Image Points:** - **Part c:** \( (3, 3) \) - **Part d:** \( (-4, -7) \)