Q22. - Algebra 2

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 $D\text{'}E\text{'}F\text{'}G\text{'}$

Step-by-Step Solution

Answer

Coordinates of the image after rotation of $270°$ counter clockwise are

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

Step 1 - Vertex matrix of parallelogram D E F G

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

Step 2 - 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

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

Step 3 - 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

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