Chapter 7.6, Problem 5CYU

To determine

Check whether the given dilationis a similarity transformation or not.

Answer to Problem 5CYU

The given dilation is a similarity transformation.

Explanation of Solution

Given:

The given dilation is

  

Calculation:

From the figure, the given vertices are

  $j\left(0,0\right),R\left(-3,1\right),S\left(-1,3\right),K\left(-6,2\right),L\left(-2,6\right)$

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

  $\begin{array}{l}JR=\sqrt{{\left(1-0\right)}^2+{\left(-3-0\right)}^2}=\sqrt{10}\\ JK=\sqrt{{\left(2-0\right)}^2+{\left(-6-0\right)}^2}=\sqrt{40}=2\sqrt{10}\end{array}$

  $\frac{JR}{JK}=\frac{\sqrt{10}}{2\sqrt{10}}=\frac{1}{2}$

  $\begin{array}{l}JS=\sqrt{{\left(3-0\right)}^2+{\left(-1-0\right)}^2}=\sqrt{10}\\ JL=\sqrt{{\left(6-0\right)}^2+{\left(-2-0\right)}^2}=\sqrt{40}=2\sqrt{10}\end{array}$

  $\frac{JS}{JL}=\frac{\sqrt{10}}{2\sqrt{10}}=\frac{1}{2}$

  $\begin{array}{l}RS=\sqrt{{\left[-1-\left(-3\right)\right]}^2+{\left(3-1\right)}^2}=\sqrt{8}=2\sqrt{2}\\ KL=\sqrt{{\left[-2-\left(-6\right)\right]}^2+{\left(6-2\right)}^2}=\sqrt{32}=4\sqrt{2}\end{array}$

  $\frac{RS}{KL}=\frac{2\sqrt{2}}{4\sqrt{2}}=\frac{1}{2}$

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

By the use of $SSS$ similarity,

  $\Delta JRS\sim \Delta JKL$ .

Hence thegiven dilation is a similarity transformation