Chapter 7.6, Problem 17PPS

To determine

Draw a graph for the original figure and its dilated image. Check whether the dilation is a similarity transformation or not.

Answer to Problem 17PPS

The dilation is a similarity transformation.

Explanation of Solution

Given:

The given vertices are

  $J\left(-6,8\right),K\left(6,6\right),L\left(-2,4\right),D\left(-12,16\right),G\left(12,12\right),H\left(-4,8\right)$

Calculation:

The graph for the given points $J\left(-6,8\right),K\left(6,6\right),L\left(-2,4\right),D\left(-12,16\right),G\left(12,12\right),H\left(-4,8\right)$ is given below.

  

Find the distance (using distance formula) of corresponding sides and find the ratio.

  $\begin{array}{l}DH=\sqrt{{\left[-12-\left(-4\right)\right]}^2+{\left[16-8\right]}^2}=\sqrt{128}=8\sqrt{2}\\ JL=\sqrt{{\left[-6-\left(-2\right)\right]}^2+{\left(8-4\right)}^2}=\sqrt{32}=4\sqrt{2}\end{array}$

  $\frac{DH}{JL}=\frac{8\sqrt{2}}{4\sqrt{2}}=2$

  $\begin{array}{l}GH=\sqrt{{\left[12-\left(-4\right)\right]}^2+{\left(12-8\right)}^2}=\sqrt{272}=4\sqrt{17}\\ KL=\sqrt{{\left[6-\left(-2\right)\right]}^2+{\left(6-4\right)}^2}=\sqrt{68}=2\sqrt{17}\end{array}$

  $\frac{GH}{KL}=\frac{4\sqrt{17}}{2\sqrt{17}}=2$

  $\begin{array}{l}DG=\sqrt{{\left[-12-12\right]}^2+{\left(16-12\right)}^2}=\sqrt{592}=4\sqrt{37}\\ JK=\sqrt{{\left[-6-6\right]}^2+{\left(8-6\right)}^2}=\sqrt{148}=2\sqrt{37}\end{array}$

  $\frac{DG}{JK}=\frac{4\sqrt{37}}{2\sqrt{37}}=2$

From the above result, the length of sides are proportional.

By the use of $SSS$ similarity,

  $\Delta JLK\sim \Delta DHG$ .

Hence the dilation is a similarity transformation.