Q19. - Algebra 2

Question

For exercises 18-20, use the following information

The vertices of $\Delta XYZ$ are $X\left(1,-1\right),Y\left(2,-4\right),\text{ and }Z\left(7,-1\right)$. The triangle is reflected over the line \[y=x\]

19. Find the coordinates of $\Delta X\text{'}Y\text{'}Z\text{'}$

Step-by-Step Solution

Answer

Coordinates of the image after a dilation by a scale factor of 3 are $X\text{'}\left(-1,1\right),Y\text{'}\left(-4,2\right),\text{ and }Z\text{'}\left(-1,7\right)$

Step 1 - Vertex matrix of Δ X Y Z

Vertex matrix of coordinates of given triangle is $\left[\begin{array}{ccc}1& 2& 7\\ -1& -4& -1\\ & & \end{array}\right]$

Step 2 - Vertex matrix after reflection

Vertex matrix of triangle after reflection over the line $y=x$  is product of matrix $\left[\begin{array}{cc}0& 1\\ 1& 0\\ & \end{array}\right]$ with vertex matrix of given triangle as shown below

$\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\times \left[\begin{array}{ccc}1& 2& 7\\ -1& -4& -1\end{array}\right]=\left[\begin{array}{ccc}-1& -4& -1\\ 1& 2& 7\end{array}\right]$

Step 3 - Coordinates of image after dilation

As, each coordinate is represented by a column of vertex matrix, so each column represents one coordinate.

Thus, coordinates of image after dilation are

$X\text{'}\left(-1,1\right),Y\text{'}\left(-4,2\right),\text{ and }Z\text{'}\left(-1,7\right)$