Chapter 9.6, Problem 40HP

To determine

To graph: A triangle on the coordinate plane then dilate the triangle so the area of dilation is 4 times the original and the scale factor.

Answer to Problem 40HP

A triangle on the coordinate plane is shown in Figure (1) and the scale factor is $2$ .

Explanation of Solution

The coordinates of the triangle are:

  $\begin{array}{c}W\left(0,2\right)\\ X\left(0,0\right)\\ Y\left(2,0\right)\end{array}$

The graph of $\Delta WXY$ is shown below.

Figure (1)

The area of triangle is calculated as,

  $\begin{array}{c}A=\frac{1}{2}\times b\times h\\ =\frac{1}{2}\times 2\times 2\\ =2\text{sq}\text{.units}\end{array}$

The area of dilation is 4 times.

  $\begin{array}{c}{A}^{\prime }=\frac{1}{2}\times W^{\prime }{X}^{\prime }\times X^{\prime }{Y}^{\prime }\\ 4\left(2\right)=\frac{1}{2}\times W^{\prime }{X}^{\prime }\times W^{\prime }{X}^{\prime }\\ {\left(W^{\prime }{X}^{\prime }\right)}^2=16\\ W^{\prime }{X}^{\prime }=4\end{array}$

And,

  $X^{\prime }{Y}^{\prime }=4$ .

The scale factor is 2.

The new coordinate are $\left(x,y\right)\to \left(kx,ky\right)$ . So,

  $\begin{array}{c}W\left(0,2\right)\to W^{\prime }\left(0,4\right)\\ X\left(0,0\right)\to X^{\prime }\left(0,0\right)\\ Y\left(2,0\right)\to Y^{\prime }\left(4,0\right)\end{array}$

The graph of the new triangle with old triangle is shown below.

Figure (2)

Therefore, the image of the polygon is shown in figure (2).