Q11. - Algebra 2

Question

A point is translated from $B$ to $C$ as shown at the right. If a point at $\left(-4,3\right)$ is translated in the same way, what will be its new coordinates?

A$\left(3,4\right)$ B$\left(1,1\right)$ C$\left(-7,8\right)$ D$\left(1,6\right)$

Step-by-Step Solution

Answer

New coordinate is $\text{B }\left(1,1\right)$

Step 1 - Vertex matrix

Let $\left(x,y\right)$ be an ordered pair then its vertex matrix will be $\left[\begin{array}{c}x\\ y\\ \end{array}\right]$

Step 2 - Find vertex matrix for points B  and  C

From image coordinates of point $B$ and $C$ are $\left(-1,3\right)$ and $\left(4,1\right)$ 

In vertex form point $\left[\begin{array}{c}-1\\ 3\\ \end{array}\right]$ is translated to point $\left[\begin{array}{c}4\\ 1\\ \end{array}\right]$

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

$\begin{array}{c}\left[\begin{array}{c}-1\\ 3\end{array}\right]+\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}4\\ 1\end{array}\right]\\ \left[\begin{array}{c}-1+x\\ 3+y\end{array}\right]=\left[\begin{array}{c}4\\ 1\end{array}\right]\end{array}$

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

$\begin{array}{c}-1+x=4\\ x=4+1\\ x=5\end{array}$and $\begin{array}{c}3+y=1\\ y=1-3\\ y=-2\end{array}$

Thus, translated matrix is

$\left[\begin{array}{c}5\\ -2\\ \end{array}\right]$

Step 4 - Find coordinate of new point

As point $\left(-4,3\right)$ with vertex matrix $\left[\begin{array}{c}-4\\ 3\\ \end{array}\right]$ is translated with same translation matrix, so sum of vertex matrix of given point with translated matrix gives vertex matrix of new point as shown below

$\begin{array}{c}\left[\begin{array}{c}-4\\ 3\end{array}\right]+\left[\begin{array}{c}5\\ -2\end{array}\right]=\left[\begin{array}{c}-4+5\\ 3-2\end{array}\right]\\ =\left[\begin{array}{c}1\\ 1\end{array}\right]\end{array}$

Thus, coordinate of new point is $\left(1,1\right)$