Perimeter of a rectangular field is 160 metres and the difference between its two adjacent sides is 48 metres. The side of a square field, having the same area as that of the rectangle, is (S.S.C. 2005)

### Concept & Area and Perimeter To find the side of the square, we first need to determine the area of the rectangle. The perimeter of a rectangle is $2(l + b)$ and its area is $l \times b$. $$ \text{Perimeter} = 2(l + b) $$ $$ \text{Area of Rectangle} = l \times b $$ $$ \text{Area of Square} = a^2 $$ ### Step-by-Step Solution * Given the perimeter of the rectangular field is 160 m: $2(l + b) = 160$, which simplifies to $l + b = 80$. * The difference between adjacent sides is 48 m: $l - b = 48$. * Adding both equations: $(l + b) + (l - b) = 80 + 48 \Rightarrow 2l = 128 \Rightarrow l = 64$ m. * Substituting $l$ back: $64 + b = 80 \Rightarrow b = 16$ m. * The area of the rectangle is $64 \times 16 = 1024$ sq m. * The area of the square is equal to the area of the rectangle: $a^2 = 1024$. * Solving for the side of the square: $a = \sqrt{1024} = 32$ m. ### Exam Strategy & Shortcut Instead of fully multiplying $64 \times 16$, you can look at the prime factorization or perfect squares. $64$ is $8^2$ and $16$ is $4^2$. Therefore, the area is $8^2 \times 4^2 = (8 \times 4)^2 = 32^2$. This immediately tells you the side of the square is 32 m, saving calculation time. ### Common Pitfall A common mistake is finding the length and breadth and then assuming one of them is the side of the square, or forgetting to take the square root of the area. ### Final Answer Therefore, the correct answer is **32 m**.