Question

A square and a rectangle have same perimeter. The length and breadth of the rectangle are 18cm and 14cm respectively. The area of the square is

• A. 269 sq.cm
• B. 144 sq.cm
• C. 169 sq.cm
• D. 256 sq.cm

Hint:

The side of a square with the same perimeter as a rectangle is always the arithmetic mean of the rectangle's length and breadth:
\[ s = \frac{l+b}{2} = \frac{18+14}{2} = 16\text{ cm} \] Square Area \( = 16^2 = 256\text{ sq. cm}\). This provides a very fast shortcut!

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
This question tests basic geometric properties of 2D shapes, specifically perimeter and area relationships between a square and a rectangle.

Step 2: Key Formulas and approach:
Let the side of the square be \(s\), and the length and breadth of the rectangle be \(l\) and \(b\).
We are given:
1. Perimeter of a rectangle:
\[ P_{\text{rect}} = 2(l + b) \] 2. Perimeter of a square:
\[ P_{\text{square}} = 4s \] Since their perimeters are equal, we equate them to solve for the square's side length \(s\).
Once \(s\) is found, we calculate the area of the square using:
\[ \text{Area}_{\text{square}} = s^2 \]

Step 3: Detailed Explanation:

• Identify the dimensions of the rectangle:
\[ l = 18\text{ cm} \] \[ b = 14\text{ cm} \]

• Compute the perimeter of the rectangle:
\[ P_{\text{rect}} = 2 \times (18 + 14) = 2 \times 32 = 64\text{ cm} \]

• Since both shapes share the same perimeter:
\[ P_{\text{square}} = 4s = 64\text{ cm} \]

• Solve for the side length of the square:
\[ s = \frac{64}{4} = 16\text{ cm} \]

• Calculate the area of the square:
\[ \text{Area}_{\text{square}} = s^2 = 16^2 = 256\text{ sq. cm} \]

Step 4: Final Answer:
The area of the square is \(256\text{ sq. cm}\), making Option (D) the correct choice.