Chapter 7.6, Problem 4CYU

To determine

Check whether the given dilationis a similarity transformation or not.

Answer to Problem 4CYU

The given dilation is a similarity transformation.

Explanation of Solution

Given:

The given dilation is

  

Calculation:

From the figure, the given vertices are

  $A\left(0,0\right),B\left(-4,-4\right),C\left(4,-2\right),D\left(-2,-2\right),E\left(2,-1\right)$

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

  $\begin{array}{l}AB=\sqrt{{\left(-4-0\right)}^2+{\left(-4-0\right)}^2}=\sqrt{32}=4\sqrt{2}\\ AD=\sqrt{{\left(-2-0\right)}^2+{\left(-2-0\right)}^2}=\sqrt{8}=2\sqrt{2}\end{array}$

  $\frac{AD}{AB}=\frac{2\sqrt{2}}{4\sqrt{2}}=\frac{1}{2}$

  $\begin{array}{l}AC=\sqrt{{\left(4-0\right)}^2+{\left(-2-0\right)}^2}=\sqrt{20}=2\sqrt{5}\\ AE=\sqrt{{\left(2-0\right)}^2+{\left(-1-0\right)}^2}=\sqrt{5}\end{array}$

  $\frac{AE}{AC}=\frac{\sqrt{5}}{2\sqrt{5}}=\frac{1}{2}$

  $\begin{array}{l}BC=\sqrt{{\left[4-\left(-4\right)\right]}^2+{\left[-2-\left(-4\right)\right]}^2}=\sqrt{68}=2\sqrt{17}\\ DE=\sqrt{{\left[2-\left(-2\right)\right]}^2+{\left[-1-\left(-2\right)\right]}^2}=\sqrt{17}\end{array}$

  $\frac{DE}{BC}=\frac{\sqrt{17}}{2\sqrt{17}}=\frac{1}{2}$

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

By the use of $SSS$ similarity,

  $\Delta ADE\sim \Delta ABC$ .

Hence thegiven dilation is a similarity transformation