Chapter 7.3, Problem 9CE

To determine

To check: Whether regular pentagons are also similar.

Answer to Problem 9CE

Yes, all regular pentagons are similar

Explanation of Solution

Given information: Polygons are regular pentagons

A pentagon is a five-sided polygon such that all its sides and all its angles are equal.

Two polygons are said to be similar if the following conditions are satisfied:

1. Corresponding angles of the polygons are equal.
2. Corresponding sides of the polygons are proportional.

First find the angle measure of each angle of a pentagon.

Number of sides of a pentagon $(n)=5$

Measure of each angle of a pentagon $=\frac{180°(n-2)}{n}$

Put $n=5$

Measure of each angle $=\frac{180°(5-2)}{5}=108°$

So, each angle of any pentagon is equal to $108°$

This means each of the corresponding angles of the pentagons are equal to $108°$ .

So, condition 1 is satisfied

Now check condition 2.

As each of the regular pentagon has equal sides, ratio of corresponding sides of the two pentagons are equal to 1.

(For example:

Consider two pentagons:

Let $a,b,c,d,e$ be the sides of one pentagon and $u,v,w,x,y$ be the sides of another pentagon.

As the pentagons are regular pentagons,

  $\begin{array}{l}a=b=c=d=e\\ u=v=w=x=y\end{array}$

This implies $\frac{a}{u}=\frac{b}{v}=\frac{c}{w}=\frac{d}{x}=\frac{e}{y}=1$

That is corresponding sides of the pentagons are proportional.

So, condition 2 is also satisfied.

Therefore, it can be concluded that all regular pentagons are similar.