Chapter 11, Problem 31CR

To determine

To find: Determine if the two figures are similar by using transformations.

Answer to Problem 31CR

Given two rectangles are similar and Rectancle PQRS is dilated with a scale factor of $k=\frac{1}{2}$ to map rectangle ABCD.

Explanation of Solution

From Given figure

We get

From rectangle $ABCD$

  $\begin{array}{l}AB=3\\ BC=2\\ CD=3\\ DA=2\end{array}$

From rectangle $PQRS$

  $\begin{array}{l}PQ=6\\ QR=4\\ RS=6\\ SP=4\end{array}$

If two polygons are similar then the ratio of the lengths of the two corresponding sides is the scale factor.

Ratio of lengths of each side are

  $\begin{array}{l}\frac{AB}{PQ}=\frac{3}{6}=\frac{1}{2}\\ \frac{BC}{QR}=\frac{2}{4}=\frac{1}{2}\\ \frac{CD}{RS}=\frac{3}{6}=\frac{1}{2}\\ \frac{AD}{SP}=\frac{2}{4}=\frac{1}{2}\end{array}$

On comparing we can say ratios are equal

So, given two rectangles are similar and Rectancle PQRS is dilated with a scale factor of $k=\frac{1}{2}$ to map rectangle ABCD.