Chapter 4.4, Problem 34PPS

To determine

To find: The coordinates of the vertices of the image of figure after the given rotation and then graph the pre-image and image.

Answer to Problem 34PPS

The coordinates of the vertices of the image are $S\text{'}\left(4,-2\right)$ , $T\text{'}\left(-1,-5\right)$ and $U\text{'}\left(2,1\right)$ , and the graph of image and pre-image is shown in the figure (1).

Explanation of Solution

Given:

The $\Delta STU$ with vertices $S\left(-4,2\right)$ , $T\left(1,5\right)$ and $U\left(-2,-1\right)$ , and rotation is $180°$ about the origin.

Calculation:

Write the vertex matrix as follows.

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

Multiply the vertex matrix by the matrix $\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]$ .

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

Therefore, the coordinates of the vertices of the image are $S\text{'}\left(4,-2\right)$ , $T\text{'}\left(-1,-5\right)$ and $U\text{'}\left(2,1\right)$ .

The graph of the image and pre-image is as follows.

  

Figure (1)