Chapter 4, Problem 12PT

To determine

To calculate: The coordinates of $\Delta XYZ\text{ where }X\left(1,2\right),Y\left(3,6\right)\text{}\text{ and }Z\left(-1,4\right)$ if it is rotated 90 degrees counterclockwise about the origin.

Answer to Problem 12PT

The new coordinates of $\Delta X\text{'}Y\text{'}Z\text{'}\text{ are }X\text{'}\left(-2,1\right),Y\text{'}\left(-6,3\right)\text{}\text{ and }Z\text{'}\left(-4,-1\right)$ .

Explanation of Solution

Given information:

The coordinates of triangle $\Delta XYZ\text{ are }X\left(1,2\right),Y\left(3,6\right)\text{}\text{ and }Z\left(-1,4\right)$ .

Calculation:

Consider the coordinates of triangle $\Delta XYZ\text{ are }X\left(1,2\right),Y\left(3,6\right)\text{}\text{ and }Z\left(-1,4\right)$ .

The vertex matrix form is provided below,

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

According to the question the triangle is rotated 90 degrees counterclockwise about the origin.

For the new coordinates of the triangle the old vertex matrix must be multiplied with $\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$

Therefore,

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

Now, new coordinates $\text{are }X\text{'}\left(-2,1\right),Y\text{'}\left(-6,3\right)\text{}\text{ and }Z\text{'}\left(-4,-1\right)$

Thus, the new coordinates of $\Delta X\text{'}Y\text{'}Z\text{'}\text{ are }X\text{'}\left(-2,1\right),Y\text{'}\left(-6,3\right)\text{}\text{ and }Z\text{'}\left(-4,-1\right)$ .