Q5. - 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.

Find the coordinates of $\Delta A\text{'}B\text{'}C\text{'}$

Step-by-Step Solution

Answer

Coordinates of $\Delta A\text{'}B\text{'}C\text{'}$ are $A\text{'}\left(4,3\right),B\text{'}\left(5,-6\right),\text{ and }C\text{'}\left(-3,-7\right)$

Step 1 - Write vertex matrix of Δ A B C

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 3 units right and 1 unit down so translation matrix is

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

Step 3 - Write vertex matrix of Δ A ' B ' C '

Sum of vertex matrix of $\Delta ABC$ with translation matrix gives vertex matrix of $\Delta A\text{'}B\text{'}C\text{'}$

$\left[\begin{array}{ccc}1& 2& -6\\ 4& -5& -6\end{array}\right]+\left[\begin{array}{ccc}3& 3& 3\\ -1& -1& -1\end{array}\right]=\left[\begin{array}{ccc}4& 5& -3\\ 3& -6& -7\end{array}\right]$

Step 4 - Find coordinates of Δ A ' B ' C '

As, each coordinate is represented by a column of vertex matrix, so each column represents one coordinate of triangle $A\text{'}B\text{'}C\text{'}$

Coordinates are $A\text{'}\left(4,3\right),B\text{'}\left(5,-6\right),\text{ and }C\text{'}\left(-3,-7\right)$