Chapter 4, Problem 5E

To determine

To calculate: The x -coordinate of the point A if the image is rotated 90 degrees counterclockwise then reflected over the line $y=x$ .

  

Answer to Problem 5E

The x -coordinate of the point Ais $2$ .

Explanation of Solution

Given information:

The image is rotated 90 degrees counterclockwise then reflected over the line $y=x$ .

  

Calculation:

Consider the image is rotated 90 degrees counterclockwise then reflected over the line $y=x$ .

  

The coordinates of polygon $DABC$ are $A\left(2,2\right),B\left(2,-2\right),C\left(-3,1\right)\text{ and }D\left(-2,5\right)$ .

The vertex matrix form is provided below,

  $\left[\begin{array}{cccc}2& 2& -3& -2\\ 2& -2& 1& 5\end{array}\right]$

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

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

Therefore,

  $\begin{array}{c}\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]\left[\begin{array}{cccc}2& 2& -3& -2\\ 2& -2& 1& 5\end{array}\right]=\left[\begin{array}{cccc}0-2& 0+2& 0-1& 0-5\\ 2+0& 2+0& -3+0& -2+0\end{array}\right]\\ =\left[\begin{array}{cccc}-2& 2& -1& -5\\ 2& 2& -3& -2\end{array}\right]\end{array}$

Now, the polygon is reflected over the line $y=x$ so multiply the matrix by $\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$ .

  $\begin{array}{c}\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\left[\begin{array}{cccc}-2& 2& -1& -5\\ 2& 2& -3& -2\end{array}\right]=\left[\begin{array}{cccc}0+2& 0+2& 0-3& 0-2\\ -2+0& 2+0& -1+0& -5+0\end{array}\right]\\ =\left[\begin{array}{cccc}2& 2& -3& -2\\ -2& 2& -1& -5\end{array}\right]\end{array}$

Now, new coordinates are $A\left(2,-2\right),B\left(2,2\right),C\left(-3,-1\right)\text{ and }D\left(-2,-5\right)$

Thus, the x -coordinate of the point Ais $2$ .