Q45. - Algebra 2

Question

Triangle $ABC$ has vertices with coordinates $A\left(-4,2\right),B\left(-4,-3\right),\text{ and }C\left(3,-2\right)$. After a dilation triangle $A\text{'}B\text{'}C\text{'}$ has coordinates $A\left(-12,6\right),B\left(-12,-9\right),\text{ and }C\left(9,-6\right)$. How many times as great is the perimeter of $A\text{'}B\text{'}C\text{'}$ as $ABC$?

 A 3  B 6  C 12   D $\frac{1}{3}$

Step-by-Step Solution

Answer

Perimeter of $A\text{'}B\text{'}C\text{'}$ is 3 times as that of $ABC$

Step 1 - Find vertex matrices for Δ A B C  and  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.

Coordinates of $\Delta ABC$ are $A\left(-4,2\right),B\left(-4,-3\right),\text{ and }C\left(3,-2\right)$, so its vertex matrix is $\left[\begin{array}{ccc}-4& -4& 3\\ 2& -3& -2\\ & & \end{array}\right]$

Again, Coordinates of $\Delta A\text{'}B\text{'}C\text{'}$ are $A\text{'}\left(-12,6\right),B\text{'}\left(-12,-9\right),\text{ and }C\text{'}\left(9,-6\right)$, so its vertex matrix is $\left[\begin{array}{ccc}-12& -12& 9\\ 6& -9& -6\\ & & \end{array}\right]$

Step 2 - Vertex matrix after dilation

Vertex matrix of triangle after dilation is product of scale factor $\left(k\right)$ with vertex matrix of given triangle as shown below

$k\times \left[\begin{array}{ccc}-4& -4& 3\\ 2& -3& -2\end{array}\right]=\left[\begin{array}{ccc}-12& -12& 9\\ 6& -9& -6\end{array}\right]$

Step 3 - Find value of scale factor

Multiply scale factor $\left(k\right)$ with vertex matrix of given triangle and equate it equal to vertex matrix of image. As both matrices are equal so corresponding elements are equal, so equate corresponding elements to get value of k as shown below

 $\begin{array}{c}\left[\begin{array}{ccc}-4k& -4k& 3k\\ 2k& -3k& -2k\end{array}\right]=\left[\begin{array}{ccc}-12& -12& 9\\ 6& -9& -6\end{array}\right]\\ -4k=-12\\ k=3\end{array}$

So, perimeter of $A\text{'}B\text{'}C\text{'}$ is 3 times as that of $ABC$