Graph and then rotate rectangle WXYZ with vertices W(-3,-5), X(1,-1),Y(3,-3) and Z(-1,-7) around the origin under the three rotations given below. You must graph the preimage and each image and identify all points from the images. (a) 90^circ counterclockwise (b) 180^circ (c) 270^circ counterclockwise

To rotate a point counterclockwise about the origin, we can use the following rotation matrix:<br /><br />$R(\theta) = \begin{pmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{pmatrix}$<br /><br />where $\theta$ is the angle of rotation in degrees.<br /><br />(a) $90^{\circ}$ counterclockwise rotation:<br /><br />$R(90^{\circ}) = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$<br /><br />Let's apply this rotation matrix to each vertex of the rectangle WXYZ:<br /><br />$W' = R(90^{\circ}) \cdot W = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \cdot \begin{pmatrix} -3 \\ -5 \end{pmatrix} = \begin{pmatrix} 5 \\ -3 \end{pmatrix}$<br /><br />$X' = R(90^{\circ}) \cdot X = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \cdot \begin{pmatrix} 1 \\ -1 \end{pmatrix} = \begin{pmatrix} 1 \\ -1 \end{pmatrix}$<br /><br />$Y' = R(90^{\circ}) \cdot Y = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \cdot \begin{pmatrix} 3 \\ -3 \end{pmatrix} = \begin{pmatrix} 3 \\ -3 \end{pmatrix}$<br /><br />$Z' = R(90^{\circ}) \cdot Z = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \cdot \begin{pmatrix} -1 \\ -7 \end{pmatrix} = \begin{pmatrix} 7 \\ -1 \end{pmatrix}$<br /><br />So, the vertices of the rotated rectangle W'X'Y'Z' are:<br /><br />$W'(5, -3), X'(1, -1), Y'(3, -3), Z'(7, -1)$<br /><br />(b) $180^{\circ}$ rotation:<br /><br />$R(180^{\circ}) = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}$<br /><br />Let's apply this rotation matrix to each vertex of the rectangle WXYZ:<br /><br />$W'' = R(180^{\circ}) \cdot W = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} \cdot \begin{pmatrix} -3 \\ -5 \end{pmatrix} = \begin{pmatrix} 3 \\ 5 \end{pmatrix}$<br /><br />$X'' = R(180^{\circ}) \cdot X = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} \cdot \begin{pmatrix} 1 \\ -1 \end{pmatrix} = \begin{pmatrix} -1 \\ 1 \end{pmatrix}$<br /><br />$Y'' = R(180^{\circ}) \cdot Y = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} \cdot \begin{pmatrix} 3 \\ -3 \end{pmatrix} = \begin{pmatrix} -3 \\ 3 \end{pmatrix}$<br /><br />$Z'' = R(180^{\circ}) \cdot Z = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} \cdot \begin{pmatrix} -1 \\ -7 \end{pmatrix} = \begin{pmatrix} 1 \\ 7 \end{pmatrix}$<br /><br />So, the vertices of the rotated rectangle W''X''Y''Z'' are:<br /><br />$W''(3, 5), X''(-1, 1), Y''(-3, 3), Z''(1, 7)$<br /><br />(c) $270^{\circ}$ counterclockwise rotation:<br /><br />$R(270^{\circ}) = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$<br /><br />Let's apply this rotation matrix to each vertex of the rectangle WXYZ:<br /><br />$W''' = R(270^{\circ}) \cdot W = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \cdot \begin{pmatrix} -3 \\ -5 \end{pmatrix} = \