Chapter 4.4, Problem 5CYP

To determine

The coordinates of the vertices of the image of triangle $XYZ$ after it is rotated $180°$ counter clockwise about the origin.

Explanation of Solution

Given Information:

The coordinates of the vertices of the triangle XYZ is

  $\begin{array}{l}X(-5,-6)\\ Y(-1,-3)\\ Z(-2,-4)\end{array}$

Calculation:

Now the vertex matrix of the triangle XYZ is,

  $\left|\begin{array}{ccc}-5& -1& -2\\ -6& -3& -4\end{array}\right|$

By multiplying the vertex matrix with rotation matrix, we get the coordinates of the reflected triangle.

Now the vertex matrix of the triangle $X\text{'}Y\text{'}Z\text{'}$ is,

  $\begin{array}{l}\left|\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right|.\left|\begin{array}{ccc}-5& -1& -2\\ -6& -3& -4\end{array}\right|=\left|\begin{array}{ccc}5& 1& 2\\ 6& 3& 4\end{array}\right|\\ \end{array}$

Thus, the coordinates of the vertices of the triangle $X\text{'}Y\text{'}Z\text{'}$ formed after rotating triangle XYZ $180°$ counter clockwise direction about the origin is,

  $\begin{array}{l}X\text{'}(5,6)\\ Y\text{'}(1,3)\\ Z\text{'}(2,4)\end{array}$