Q2. - Algebra 2

Question

Write the translation matrix for $\Delta ABC$ and its image $\Delta A\text{'}B\text{'}C\text{'}$ shown at the right.

Step-by-Step Solution

Answer

The translation matrix is $\left[\begin{array}{ccc}-3& -3& -3\\ -2& -2& -2\\ & & \end{array}\right]$

Step 1 - Write general translation matrix

Since, an ordered pair is represented by a column matrix 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 2 - Find vertex matrices for Δ A B C  and  Δ A ' B ' C '

Coordinates of $\Delta ABC$ are $A\left(-2,3\right),B\left(4,0\right),\text{ and }C\left(2,-5\right)$, so its vertex matrix is

$\left[\begin{array}{ccc}-2& 4& 2\\ 3& 0& -5\\ & & \end{array}\right]$

Again, Coordinates of $\Delta A\text{'}B\text{'}C\text{'}$ are $A\text{'}\left(-5,1\right),B\text{'}\left(1,-2\right),\text{ and }C\text{'}\left(-1,-7\right)$, so its vertex matrix is 

$\left[\begin{array}{ccc}-5& 1& -1\\ 1& -2& -7\\ & & \end{array}\right]$

Step 3 - Find translated matrix

When translation occurs, sum of vertex matrix of pre-image and translated matrix gives vertex matrix of image as shown below

 $\begin{array}{c}\left[\begin{array}{ccc}-2& 4& 2\\ 3& 0& -5\end{array}\right]+\left[\begin{array}{ccc}x& x& x\\ y& y& y\end{array}\right]=\left[\begin{array}{ccc}-5& 1& -1\\ 1& -2& -7\end{array}\right]\\ \left[\begin{array}{ccc}-2+x& 4+x& 2+x\\ 3+y& 0+y& -5+y\end{array}\right]=\left[\begin{array}{ccc}-5& 1& -1\\ 1& -2& -7\end{array}\right]\end{array}$

As elements of both matrices are equal find values of x and y

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

Thus, translated matrix is

$\left[\begin{array}{ccc}-3& -3& -3\\ -2& -2& -2\end{array}\right]$