Q13. - Algebra 2

Question

Triangle $DEF$ with vertices $D\left(1,4\right),\text{ E}\left(2,-5\right),\text{ and }F\left(-6,-6\right)$ is translated 4 units left and 2 units up.

13. Find the coordinates of $\Delta D\text{'}E\text{'}F\text{'}$

Step-by-Step Solution

Answer

Coordinates of $\Delta D\text{'}E\text{'}F\text{'}$ are $D\text{'}\left(-3,6\right),E\text{'}\left(-2,-3\right),\text{ and }F\text{'}\left(-10,-4\right)$

Step 1 - Write vertex matrix of Δ D E F

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.

First row represent $x-\text{coordinate}$ and second row represents $y-\text{coordinate}$ of each vertex.

Thus vertex matrix is $\left[\begin{array}{ccc}1& 2& -6\\ 4& -5& -6\\ & & \end{array}\right]$

Step 2 - Write translation matrix

As the triangle is translated 4 units left and 2 units up so translation matrix is

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

Step 3 - Write vertex matrix of Δ D ' E ' F '

Sum of vertex matrix of $\Delta DEF$ with translation matrix gives vertex matrix of $\Delta D\text{'}E\text{'}F\text{'}$

$\left[\begin{array}{ccc}1& 2& -6\\ 4& -5& -6\end{array}\right]+\left[\begin{array}{ccc}-4& -4& -4\\ 2& 2& 2\end{array}\right]=\left[\begin{array}{ccc}-3& -2& -10\\ 6& -3& -4\end{array}\right]$

Step 4 - Find coordinates of Δ D ' E ' F '

As, each coordinate is represented by a column of vertex matrix, so each column represents one coordinate of triangle $\Delta D\text{'}E\text{'}F\text{'}$

Coordinates are $D\text{'}\left(-3,6\right),E\text{'}\left(-2,-3\right),\text{ and }F\text{'}\left(-10,-4\right)$