Chapter 9, Problem 21CLT

To determine

To find: the vertices of the image of the square with the vertices $\left(0,2\right)$ , $\left(0,5\right)$ , $\left(3,2\right)$ and $\left(3,5\right)$ after the transformation; reflection in the x -axis

Answer to Problem 21CLT

  $\left(0,-2\right)$ , $\left(0,-5\right)$ , $\left(3,-2\right)$ and $\left(3,-5\right)$

Explanation of Solution

First write the vertices $\left(0,2\right)$ , $\left(0,5\right)$ , $\left(3,2\right)$ and $\left(3,5\right)$ as column matrices as shown below:

  $\left[\begin{array}{c}0\\ 2\end{array}\right]$ , $\left[\begin{array}{c}0\\ 5\end{array}\right]$ , $\left[\begin{array}{c}3\\ 2\end{array}\right]$ and $\left[\begin{array}{c}3\\ 5\end{array}\right]$

Now, the transformation matrix for reflection in the x -axis is $\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$ .

Now, one by one multiply all the column matrices obtained above by the matrix $\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$ .

  $\begin{array}{c}\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]\left[\begin{array}{c}0\\ 2\end{array}\right]=\left[\begin{array}{c}0+0\\ 0+\left(-2\right)\end{array}\right]\\ =\left[\begin{array}{c}0\\ -2\end{array}\right]\end{array}$

  $\begin{array}{c}\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]\left[\begin{array}{c}0\\ 5\end{array}\right]=\left[\begin{array}{c}0+0\\ 0+\left(-5\right)\end{array}\right]\\ =\left[\begin{array}{c}0\\ -5\end{array}\right]\end{array}$

  $\begin{array}{c}\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]\left[\begin{array}{c}3\\ 2\end{array}\right]=\left[\begin{array}{c}3+0\\ 0+\left(-2\right)\end{array}\right]\\ =\left[\begin{array}{c}3\\ -2\end{array}\right]\end{array}$

  $\begin{array}{c}\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]\left[\begin{array}{c}3\\ 5\end{array}\right]=\left[\begin{array}{c}3+0\\ 0+\left(-5\right)\end{array}\right]\\ =\left[\begin{array}{c}3\\ -5\end{array}\right]\end{array}$

So, the vertices of the image of the given square are $\left(0,-2\right)$ , $\left(0,-5\right)$ , $\left(3,-2\right)$ and $\left(3,-5\right)$ .