Q24. - Algebra 2

Question

Triangle $DEF$ with vertices $D\left(-2,2\right),E\left(3,5\right),\text{ and }F\left(5,-2\right)$ is translated so that $D\text{'}$ is at $\left(1,-5\right)$. Find the coordinates of $E\text{'}\text{ and }F\text{'}$. 

Step-by-Step Solution

Answer

Coordinates are $E\text{'}\left(4,8\right)\text{ and }F\text{'}\left(6,1\right)$

Step 1 - Vertex matrix of triangle D E F

Vertex matrix of coordinates of given triangle is$\left[\begin{array}{ccc}-2& 3& 5\\ 2& 5& -2\\ & & \end{array}\right]$

Step 2 - Translation matrix

Since, an ordered pair is represented by a column matrix $\left[\begin{array}{c}x\\ y\\ \end{array}\right]$ and triangle has 3 ordered pair on vertices so required matrix has 2 rows and 3 columns.

If the matrix is translated to $x$ units right or left then $x$ is added or subtracted from first row respectively.

If the matrix is translated to $y$ units up or down then $y$ is added or subtracted from second row respectively.

Thus translation matrix is of the form

$\left[\begin{array}{ccc}x& x& x\\ y& y& y\\ & & \end{array}\right]$

Step 3 - Vertex matrix of triangle D ' E ' F '

Let the coordinates of $E\text{'}$ be $\left(x_1,y_1\right)$ and coordinates of $F\text{'}$  be $\left(x_2,y_2\right)$ 

So vertex matrix is $\left[\begin{array}{ccc}-1& x_1& x_2\\ 5& y_1& y_2\\ & & \end{array}\right]$ But vertex matrix of triangle after translation is sum of vertex matrix of original triangle with translated matrix as shown below

 $\begin{array}{c}\left[\begin{array}{ccc}-2& 3& 5\\ 2& 5& -2\end{array}\right]+\left[\begin{array}{ccc}x& x& x\\ y& y& y\end{array}\right]=\left[\begin{array}{ccc}-1& x_1& x_2\\ 5& y_1& y_2\end{array}\right]\\ \left[\begin{array}{ccc}-2+x& 3+x& 5+x\\ 2+y& 5+y& -2+y\end{array}\right]=\left[\begin{array}{ccc}-1& x_1& x_2\\ 5& y_1& y_2\end{array}\right]\end{array}$

Step 4 - Find values of x  and  y

Since, two matrices are equal, so corresponding elements are equal

Equating elements find values of $x\text{ and }y$

$\begin{array}{c}-2+x=-1\\ x=-1+2\\ x=1\end{array}$  and  $\begin{array}{c}2+y=5\\ y=5-2\\ y=3\end{array}$

Step 5 - Find coordinates of E '  and  F '

Substitute 1 for x and 3 for y to get coordinates of $E\text{'}$ and $F\text{'}$

$\begin{array}{c}\left[\begin{array}{ccc}-1& x_1& x_2\\ 5& y_1& y_2\end{array}\right]=\left[\begin{array}{ccc}-1& 3+1& 5+1\\ 5& 5+3& -2+3\end{array}\right]\\ =\left[\begin{array}{ccc}-1& 4& 6\\ 5& 8& 1\end{array}\right]\end{array}$

Thus, coordinates of $E\text{'}$ is $\left(4,8\right)$ and $F\text{'}$ is $\left(6,1\right)$