Chapter 7.6, Problem 14PPS

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 14PPS

The dilation is a similarity transformation.

Explanation of Solution

Given:

The given vertices are

  

Calculation:

The graph for the given points $M\left(1,4\right),P\left(2,2\right),Q\left(5,5\right),S\left(-3,6\right),T\left(0,0\right),U\left(9,9\right)$ is given below.

  

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

  $\begin{array}{l}ST=\sqrt{{\left(-3-0\right)}^2+{\left(6-0\right)}^2}=\sqrt{45}=3\sqrt{5}\\ MP=\sqrt{{\left(1-2\right)}^2+{\left(4-2\right)}^2}=\sqrt{5}\end{array}$

  $\frac{MP}{ST}=\frac{\sqrt{5}}{3\sqrt{5}}=\frac{1}{3}$

  $\begin{array}{l}SU=\sqrt{{\left(-3-9\right)}^2+{\left(6-9\right)}^2}=\sqrt{153}=3\sqrt{17}\\ MQ=\sqrt{{\left(1-5\right)}^2+{\left(4-5\right)}^2}=\sqrt{17}\end{array}$

  $\frac{MQ}{SU}=\frac{\sqrt{17}}{3\sqrt{17}}=\frac{1}{3}$

  $\begin{array}{l}TU=\sqrt{{\left(0-9\right)}^2+{\left(0-9\right)}^2}=\sqrt{162}=9\sqrt{2}\\ PQ=\sqrt{{\left(2-5\right)}^2+{\left(2-5\right)}^2}=3\sqrt{2}\end{array}$

  $\frac{PQ}{TU}=\frac{3\sqrt{2}}{9\sqrt{2}}=\frac{1}{3}$

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

By the use of $SSS$ similarity,

  $\Delta MPQ\sim \Delta STU$ .

Hence the dilation is a similarity transformation.