Question

Sum of the areas of two squares is 400 cm². If the difference of their perimeters is 16 cm, find the sides of the two squares. 

Answer 1

Let the sides of two squares by a and b respectively.
Then, area of one square, S₁ = a²
And, area of second square, S₂ = b²
Given, S₁ + S₂ = 400 cm²
⇒ a² + b² = 400 cm² …..(1)
Also, difference in perimeter = 16 cm
⇒ 4a - 4b = 16 cm
⇒ a - b = 4
⇒ a = (4 + b)

Substituting the value of 'a' in (1), we get
(4 + b)² + b² = 400
⇒ 16 + 8b + b² + b² = 400
⇒ 2b² + 8b - 384 = 0
⇒ b² + 4b - 192 = 0
⇒ b² + 16b - 12b - 192 = 0
⇒ b(b + 16) - 12(b + 16) = 0
⇒ (b +16)(b - 12) = 0
⇒ b + 16 = 0 or b - 12 = 0
⇒ b = - 16 or b = 12

Since, the side of a square cannot be negative, we reject - 16.
Thus, b = 12
⇒ a = 4 + b = 4 + 12 = 16

Hence, the sides of a square are 16 cm and 12 cm respectively.