Q25. - Algebra 1

Question

In each figure, only one side of each regular pentagon is shared with another pentagon. Each side of each pentagon is 1 centimeter. If the pattern continues, what is the perimeter of a figure that has 6 pentagons?

F 15 cm

H 20 cm

G 25 cm

J 30 cm

Step-by-Step Solution

Answer

The perimeter of a figure having 6 pentagons is option H 20 cm.

Step 1. State the concept of perimeter.

In geometry, perimeter can be defined as the path or the boundary that surrounds a shape. It can also be defined as the length of the outline of a shape.

Step 2. Find the perimeters of the pentagons given in the figure.

Perimeter of the figure with 1 pentagon is,

$$\begin{gathered} P_1=5\times 1 \\ =5\text{\hspace{0.17em}\hspace{0.17em}cm} \end{gathered}$$ 

Perimeter of the figure with 2 pentagons is,

$$\begin{gathered} P_2=\left(4\times 1\right)+\left(4\times 1\right) \\ =4+4 \\ =8\text{\hspace{0.17em}cm} \end{gathered}$$

Perimeter of the figure with 3 pentagons is,

 $$\begin{gathered} P_3=\left(4\times 1\right)+\left(4\times 1\right)+\left(3\times 1\right) \\ =4+4+3 \\ =11\text{\hspace{0.17em} cm} \end{gathered}$$

Therefore, the perimeters are 5 cm, 8 cm and 11 cm respectively.

Step 3. Identify the pattern in which the perimeters are changing with the increase of 1 pentagon in each figures.

First find the difference between the perimeters of corresponding figures.

 $\begin{array}{l}d=P_2-P_1=8-5=3\\ d=P_3-P_2=11-8=3\end{array}$ 

Notice that the common difference is constant. Therefore the sequence of perimeters are in Arithmetic progression.

Step 4. Find the general formula for the perimeters of n polygons.

The perimeter of n polygon $\left(P_n\right)$ is given by

$P_n=P_1+\left(n-1\right)d$ 

Where d is the common difference.

Here $P_{1.}=5$ and $d=3$

$$\begin{gathered} P_{n.}=5+\left(n-1\right)3 \\ =5+3n-3 \\ {P}_{n.}=3n+2\dots \left(1\right) \end{gathered}$$ 

Step 5. Calculate the perimeter of 6 pentagons.

In order to calculate the perimeter of 6 pentagons substitute $n=6$ in equation

$$\begin{gathered} P_6=3\left(6\right)+2 \\ =18+2 \\ {P}_6=20 \end{gathered}$$

Therefore, the perimeter of the figure that has 6 pentagons is 20 cm.

 

Option H is correct.