Q41. - Algebra 2

Question

FOOTPRINTS

For exercises 38-41, use the following information.

The combination of a reflection and a translation is called a glide reflection.

An example is a set of footprints.

41. What are the coordinates of next two foot prints?

Step-by-Step Solution

Answer

Coordinates of next two foot prints are $C\left(17,-2\right)\text{ and }D\left(23,2\right)$

Step 1 - Vertex matrix of a point

An ordered pair $\left(x,y\right)$ is represented by a column matrix $\left[\begin{array}{c}x\\ y\\ \end{array}\right]$ which is known as vertex matrix of an ordered pair.

For the coordinates $A\left(5,-2\right)$ and $B\left(11,2\right)$, vertex matrices are $\left[\begin{array}{c}5\\ -2\\ \end{array}\right]\text{ and }\left[\begin{array}{c}11\\ 2\\ \end{array}\right]$ respectively.

For the coordinates $C\left(x_1,y_1\right)\text{ and }D\left(x_2,y_2\right)$, let the vertex matrix be $\left[\begin{array}{c}{x}_1\\ y_1\\ \end{array}\right]\text{ and }\left[\begin{array}{c}{x}_2\\ y_2\\ \end{array}\right]$ respectively.

Step 2 - Coordinate of point C

Multiply matrix $\left[\begin{array}{cc}1& 0\\ 0& -1\\ & \end{array}\right]$ with vertex matrix of B, then add translated matrix $\left[\begin{array}{c}6\\ 0\\ \end{array}\right]$ gives vertex matrix $\left[\begin{array}{c}{x}_1\\ y_1\\ \end{array}\right]$ as shown below

 $\begin{array}{c}\left[\begin{array}{c}{x}_1\\ y_1\end{array}\right]=\left(\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]\times \left[\begin{array}{c}11\\ 2\end{array}\right]\right)+\left[\begin{array}{c}6\\ 0\end{array}\right]\\ =\left[\begin{array}{c}11\\ -2\end{array}\right]+\left[\begin{array}{c}6\\ 0\end{array}\right]\\ =\left[\begin{array}{c}11+6\\ -2+0\end{array}\right]\\ =\left[\begin{array}{c}17\\ -2\end{array}\right]\end{array}$

So coordinate of point $C$ is $\left(17,-2\right)$

Step 3 - Coordinate of point D

Multiply matrix $\left[\begin{array}{cc}1& 0\\ 0& -1\\ & \end{array}\right]$ with vertex matrix of $C$, then add translated matrix $\left[\begin{array}{c}6\\ 0\\ \end{array}\right]$ gives vertex matrix $\left[\begin{array}{c}{x}_2\\ y_2\\ \end{array}\right]$ as shown below

 $\begin{array}{c}\left[\begin{array}{c}{x}_2\\ y_2\end{array}\right]=\left(\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]\times \left[\begin{array}{c}17\\ -2\end{array}\right]\right)+\left[\begin{array}{c}6\\ 0\end{array}\right]\\ =\left[\begin{array}{c}17\\ 2\end{array}\right]+\left[\begin{array}{c}6\\ 0\end{array}\right]\\ =\left[\begin{array}{c}17+6\\ 2+0\end{array}\right]\\ =\left[\begin{array}{c}23\\ 2\end{array}\right]\end{array}$

So coordinate of point C is $\left(23,2\right)$