Chapter 2.4, Problem 27E

To determine

To find: coordinates of the composite transformation and then graph the pre-image and image on the same coordinate grid.

Answer to Problem 27E

  $\left(6,-4\right),\left(3,-2\right)$ and $\left(1,2\right)$

  

Explanation of Solution

Given information:

The vertices of $\Delta JKL$ are $J\left(-6,4\right),K\left(-3,2\right)$ and $L\left(-1,-2\right)$ .

The composite transition if $R_{y-\text{axis}}\circ R_{x-\text{axis}}$ .

Calculation:

The matrix for $R_{x-\text{axis}},R_{y-\text{axis}}\text{ and vertex matrix for }\Delta JKL$ is given below.

  $\begin{array}{l}{R}_{x-\text{axis}}=\left[\begin{array}{cc}\begin{array}{l}0\\ -1\end{array}& \begin{array}{l}1\\ 0\end{array}\end{array}\right]\\ R_{y-\text{axis}}=\left[\begin{array}{cc}\begin{array}{l}-1\\ 0\end{array}& \begin{array}{l}0\\ 1\end{array}\end{array}\right]\\ {\text{V}}_{\Delta JKL}=\left[\begin{array}{ccc}\begin{array}{l}-6\\ 4\end{array}& \begin{array}{l}-3\\ 2\end{array}& \begin{array}{l}-1\\ -2\end{array}\end{array}\right]\end{array}$

Solve $R_{y-\text{axis}}\circ R_{x-\text{axis}}$ as follows:

  $\begin{array}{c}{R}_{y-\text{axis}}\circ R_{x-\text{axis}}=\left[\begin{array}{cc}\begin{array}{l}-1\\ 0\end{array}& \begin{array}{l}0\\ 1\end{array}\end{array}\right]\left[\begin{array}{cc}\begin{array}{l}0\\ -1\end{array}& \begin{array}{l}1\\ 0\end{array}\end{array}\right]\\ =\left[\begin{array}{cc}\begin{array}{l}-1\\ 0\end{array}& \begin{array}{l}0\\ -1\end{array}\end{array}\right]\end{array}$... (1)

Now multiply matrix (1) with vertex matrix of the given triangle as follows:

  $\begin{array}{c}\left[\begin{array}{cc}\begin{array}{l}-1\\ 0\end{array}& \begin{array}{l}0\\ -1\end{array}\end{array}\right]\left[\begin{array}{ccc}\begin{array}{l}-6\\ 4\end{array}& \begin{array}{l}-3\\ 2\end{array}& \begin{array}{l}-1\\ -2\end{array}\end{array}\right]=\left[\begin{array}{ccc}\begin{array}{l}6+0\\ 0-4\end{array}& \begin{array}{l}3+0\\ 0-2\end{array}& \begin{array}{l}1+0\\ 0+2\end{array}\end{array}\right]\\ =\left[\begin{array}{ccc}\begin{array}{l}6\\ -4\end{array}& \begin{array}{l}3\\ -2\end{array}& \begin{array}{l}1\\ 2\end{array}\end{array}\right]\end{array}$... (2)

The coordinates of the triangle $\left(\text{let's say, }\Delta J^{\text{'}}{K}^{\text{'}}{L}^{\text{'}}\right)$ by composite translation is $\left(6,-4\right),\left(3,-2\right)$ and $\left(1,2\right)$ .

The graph of the original (black color) and the dilated (red color) triangles is as follows: