A square of area `100 cm^(2)`  has the same perimeter as of a regular pentagon. What is the length of the side of the regular pentagon?

Options

A. 6 cm
B. 8 cm
C. 7 cm
D. 5 cm

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Find the side length of the square.
Given that the area of the square is \(100 \, \text{cm}^2\), we can use the formula for the area of a square:
\[ \text{Area} = \text{side}^2 \]
Let the side length of the square be \(x\). Therefore, we have:
\[ x^2 = 100 \]
Taking the square root of both sides gives:
\[ x = \sqrt{100} = 10 \, \text{cm} \]

### Step 2: Calculate the perimeter of the square.
The perimeter \(P\) of a square is given by the formula:
\[ P = 4 \times \text{side} \]
Substituting the value of the side we found:
\[ P = 4 \times 10 = 40 \, \text{cm} \]

### Step 3: Set the perimeter of the pentagon equal to the perimeter of the square.
Since the perimeter of the square is the same as the perimeter of the regular pentagon, we have:
\[ \text{Perimeter of pentagon} = 40 \, \text{cm} \]

### Step 4: Express the perimeter of the pentagon in terms of its side length.
Let the side length of the regular pentagon be \(y\). The perimeter of a regular pentagon (which has 5 equal sides) is given by:
\[ \text{Perimeter} = 5 \times y \]
Setting this equal to the perimeter of the square, we have:
\[ 5y = 40 \]

### Step 5: Solve for \(y\).
To find the length of one side of the pentagon, divide both sides of the equation by 5:
\[ y = \frac{40}{5} = 8 \, \text{cm} \]

### Conclusion:
The length of the side of the regular pentagon is \(8 \, \text{cm}\).

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