Q33. - Algebra 2

Question

LANDSCAPING

For exercises 33 and 34, use the following information.

A garden design is plotted on a coordinated grid. The original plan shows a fountain with vertices at $\left(-2,-2\right),\left(-6,-2\right),\left(-8,-5\right),\text{ and }\left(-4,-5\right)$. Changes to the plan now require that the fountain’s perimeter be the three-fourths that of the original.

33. Determine the new coordinates for the fountain.

Step-by-Step Solution

Answer

New coordinates for the fountain are 

$\left(-\frac{3}{2},-\frac{3}{2}\right),\left(-\frac{9}{2},-\frac{3}{2}\right),\left(-6,-\frac{15}{4}\right),\text{ and }\left(-3,-\frac{15}{4}\right)$

Step 1 - Vertex matrix

An ordered pair $\left(x,y\right)$ is represented by a column matrix $\left[\begin{array}{c}x\\ y\\ \end{array}\right]$ which is known as vertex matrix of an ordered pair

Since, a quadrilateral has 4 ordered pair on vertices so required matrix has 2 rows and 4 columns.

First row represent $x-\text{coordinate}$ and second row represents $y-\text{coordinate}$ of each vertex.

As coordinates of the vertices of quadrilateral are $\left(-2,-2\right),\left(-6,-2\right),\left(-8,5\right),\text{ and }D\left(-4,-5\right)$

Write each ordered pair as a column of a matrix to get

$\left[\begin{array}{cccc}-2& -6& -8& -4\\ -2& -2& -5& -5\\ & & & \end{array}\right]$

Step 2 - Vertex matrix after dilation

As perimeter is reduced to three-fourth of original quadrilateral, so vertex matrix of image is formed after dilation.

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

$\frac{3}{4}\times \left[\begin{array}{cccc}-2& -6& -8& -4\\ -2& -2& -5& -5\end{array}\right]=\left[\begin{array}{cccc}-\frac{3}{2}& -\frac{9}{2}& -6& -3\\ -\frac{3}{2}& -\frac{3}{2}& -\frac{15}{4}& -\frac{15}{4}\end{array}\right]$

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

 $\left(-\frac{3}{2},-\frac{3}{2}\right),\left(-\frac{9}{2},-\frac{3}{2}\right),\left(-6,-\frac{15}{4}\right),\text{ and }\left(-3,-\frac{15}{4}\right)$