Transformation of Graphs Using Matrices - Rotations

Example:

Find the coordinates of the vertices of the image $\Delta XYZ$ with $X(1,2),Y(3,5)\text{and}Z(-3,4)$ after it is rotated $180°$ counterclockwise about the origin.

Write the ordered pairs as a vertex matrix.

$\left[\begin{array}{ccc}\hfill 1& \hfill 3& \hfill -3\\ \hfill 2& \hfill 5& \hfill 4\end{array}\right]$

To rotate the $\Delta XYZ$ 180° counterclockwise about the origin, multiply the vertex matrix by the rotation matrix, $\left[\begin{array}{cc}\hfill -1& \hfill 0\\ \hfill 0& \hfill -1\end{array}\right]$ .

$\left[\begin{array}{cc}\hfill -1& \hfill 0\\ \hfill 0& \hfill -1\end{array}\right]\cdot \left[\begin{array}{ccc}\hfill 1& \hfill 3& \hfill -3\\ \hfill 2& \hfill 5& \hfill 4\end{array}\right]=\left[\begin{array}{ccc}\hfill -1& \hfill -3& \hfill 3\\ \hfill -2& \hfill -5& \hfill -4\end{array}\right]$

Therefore, the coordinates of the vertices of $\Delta X^{\text{'}}{Y}^{\text{'}}{Z}^{\text{'}}$ are $X^{\text{'}}(-1,-2),Y^{\text{'}}(-3,-5),\text{and}{Z}^{\text{'}}(3,-4)$ .

Notice that the image $\Delta X^{\text{'}}{Y}^{\text{'}}{Z}^{\text{'}}$ is congruent to the preimage $(\Delta XYZ)$ . Both figures have the same size and same shape.