Q57. - Algebra 2

Question

For exercises 56-58, use the following information

The vertices of $\Delta ABC$ are $A\left(-2,1\right),B\left(1,2\right),\text{ and }C\left(2,-3\right)$. The triangle is dilated so that its perimeter is $2\frac{1}{2}$  times the original perimeter.

57. 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 $2\frac{1}{2}$ are $A\text{'}\left(-5,2\frac{1}{2}\right),B\text{'}\left(2\frac{1}{2},5\right),\text{ and }C\text{'}\left(5,-7\frac{1}{2}\right)$

Step 1 - Vertex matrix of Δ A B C

Vertex matrix of coordinates of given triangle is $\left[\begin{array}{ccc}-2& 1& 2\\ 1& 2& -3\\ & & \end{array}\right]$

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

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

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

$A\text{'}\left(-5,2\frac{1}{2}\right),B\text{'}\left(2\frac{1}{2},5\right),\text{ and }C\text{'}\left(5,-7\frac{1}{2}\right)$