Question

For exercises 21-23, use the following information

Parallelogram $DEFG$ with $D\left(2,4\right),E\left(5,4\right),F\left(4,1\right),\text{and}G\left(1,1\right)$ is rotated $270°$ counter clockwise about the origin.

22. Find the coordinates of parallelogram $D\text{'}E\text{'}F\text{'}G\text{'}$

Short answer

Coordinates of the image after rotation of $270°$counter clockwiseare

$D\text{'}\left(4,-2\right),E\text{'}\left(4,-5\right),F\text{'}\left(1,-4\right),\text{and}G\text{'}\left(1,-1\right)$

Step-by-step solution

- Vertex matrix of parallelogram DEFG

Vertex matrix of coordinates of given parallelogram is $\left[\begin{array}{cccc}2& 5& 4& 1\\ 4& 4& 1& 1\\ & & & \end{array}\right]$

- Vertex matrix after rotation

Vertex matrix of parallelogram after rotation of $270°$counter clockwise is product of matrix $\left[\begin{array}{cc}0& 1\\ -1& 0\\ & \end{array}\right]$with vertex matrix of given parallelogram as shown below

$\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]\times \left[\begin{array}{cccc}2& 5& 4& 1\\ 4& 4& 1& 1\end{array}\right]=\left[\begin{array}{cccc}4& 4& 1& 1\\ -2& -5& -4& -1\end{array}\right]$

- Coordinates of image after rotation

As, each coordinate is represented by a column of vertex matrix, so each column represents one coordinate.

Thus, coordinates of image after rotation are

$D\text{'}\left(4,-2\right)\text{,}E\text{'}\left(4,-5\right)\text{,}F\text{'}\left(1,-4\right)\text{, and}G\text{'}\left(1,-1\right)$