Q16. - Algebra 2

Question

For exercises 15-17, use the following information

The vertices of $\Delta ABC$ are $A\left(0,2\right),B\left(1.5,-1.5\right),\text{ and }C\left(-2.5,0\right)$. The triangle is dilated so that its perimeter is three times the original perimeter.

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

Step-by-Step Solution

Answer

Coordinates of the image after a dilation by a scale factor of 3 are $A\text{'}\left(0,6\right),B\text{'}\left(4.5,-4.5\right),\text{ and }C\text{'}\left(-7.5,0\right)$

Step 1 - Vertex matrix of Δ A B C

Vertex matrix of coordinates of given triangle is $\left[\begin{array}{ccc}0& 1.5& -2.5\\ 2& -1.5& 0\end{array}\right]$

Step 2 - Vertex matrix after dilation

Vertex matrix of triangle after dilation is product of scale factor with vertex matrix of given triangle as shown below

$3\times \left[\begin{array}{ccc}0& 1.5& -2.5\\ 2& -1.5& 0\end{array}\right]=\left[\begin{array}{ccc}0& 4.5& -7.5\\ 6& -4.5& 0\end{array}\right]$

Step 3 - Coordinates of image after dilation

As, each coordinate is represented by a column of vertex matrix, so each column represents one coordinate.

Thus, coordinates of image after dilation are

$A\text{'}\left(0,6\right),B\text{'}\left(4.5,-4.5\right),\text{ and }C\text{'}\left(-7.5,0\right)$