Question

A rectangle and a square have the same perimeter. What is the magnitude of difference in their area? (a and b are the sides of the rectangle)

1. (a + b)²
2. (a - b)²
3. ((a + b)/2)²
4. ((a - b)/2)²

Answer 1

Correct Answer - Option 4 : ((a - b)/2)²

Calculation:

a and b are the sides of rectangle and let c be the side of the square,

2 × (a + b) = 4c

⇒ a + b = 2c

⇒ c = (a + b)/2 

Difference in area

⇒ c² - ab = ((a + b)/2)² - ab

⇒ ((a + b)² - 4ab)/4

⇒ (a - b)²/4

⇒ ((a - b)/2)²

∴ The magnitude of difference in their area is ((a - b)/2)2.

(a + b)² = a² + b² + 2ab

(a - b)2 = a2 + b2 - 2ab