Q4. - Algebra 2

Question

Triangle $ABC$ with vertices $A\left(1,4\right),B\left(2,-5\right),\text{ and }C\left(-6,-6\right)$ is translated 3 units right and 1 unit down.

4. Write the translation matrix.

Step-by-Step Solution

Answer

The translation matrix is $\left[\begin{array}{ccc}3& 3& 3\\ -1& -1& -1\\ & & \end{array}\right]$

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 value of x  and  y

As the triangle is translated 3 units right, so $x=3$

As the triangle is translated 1 unit down, so $y=-1$ 

Step 3 - Find translated matrix

Substitute 3 for x and $-1$ for y in general translated matrix gives required matrix

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