Chapter 12.5, Problem 42PPSE

Solution

To find: The coordinates of image after the reflection across the line.

B. $X\text{'}\left(1,5\right),Y\text{'}\left(-3,-5\right),Z\text{'}\left(3,-1\right)$

Given information:

Given, coordinates of $X\left(5,1\right),Y\left(-5,-3\right),Z\left(-1,3\right)$ are reflected across the line $y=x$ .

Calculation:

The coordinates in the matrix form is given by:

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

The reflection of a figure across $y=x$ is given by multiplying the matrix with:

  $\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$

The coordinates after the reflection are given by:

  $\begin{array}{l}\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\left[\begin{array}{ccc}5& -5& -1\\ 1& -3& 3\end{array}\right]=\left[\begin{array}{ccc}\left(0\right)\left(5\right)+\left(1\right)\left(1\right)& \left(0\right)\left(-5\right)+\left(1\right)\left(-3\right)& \left(0\right)\left(-1\right)+\left(1\right)\left(3\right)\\ \left(1\right)\left(5\right)+\left(0\right)\left(1\right)& \left(1\right)\left(-5\right)+\left(0\right)\left(-3\right)& \left(1\right)\left(-1\right)+\left(0\right)\left(3\right)\end{array}\right]\\ \left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\left[\begin{array}{ccc}5& -5& -1\\ 1& -3& 3\end{array}\right]=\left[\begin{array}{ccc}0+1& 0-3& 0+3\\ 5+0& -5+0& -1+0\end{array}\right]\\ \left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\left[\begin{array}{ccc}5& -5& -1\\ 1& -3& 3\end{array}\right]=\left[\begin{array}{ccc}1& -3& 3\\ 5& -5& -1\end{array}\right]\end{array}$

The coordinates after reflection are:

  $X\text{'}\left(1,5\right),Y\text{'}\left(-3,-5\right),Z\text{'}\left(3,-1\right)$

Conclusion:

B. $X\text{'}\left(1,5\right),Y\text{'}\left(-3,-5\right),Z\text{'}\left(3,-1\right)$