In the following question, three statements are given. You have to find which is/are necessary and sufficient to answer the following question. If a square, a rectangle, and a circle have the same perimeter. Find which figure will have the maximum area enclosed. Statement 1∶ If the side of the square is A, length of the rectangle is L, and the radius of the circle is R. A ≠ L Statement 2∶ Ratio of length and breadth of Rectangle = 5 ∶ 2. Statement 3∶ If an equilateral triangle is inscribed in the circle and the side of the triangle is 5 cm.?

Option 4 : None of these statements is required to answer the question.

The perimeter of the square, the rectangle, and the circle is same

⇒ 4A = 2(L + B) = 2πr

The area of rectangle will always be less than the area of the square.

This can be verified by taking an example,

Put A = 8

Area of square = 64 cm²

⇒ 32 = 2(L + B)

⇒ (L + B) = 16

The maximum area possible for the rectangle will be when L = B = 8 (which can’t be taken otherwise it will be a square)

∴ The area of the rectangle will always be less than the area of the square

When A = 8

⇒ 4a = 2πr

⇒ 32 = 2 × 22/7 × r

⇒ r = 56/11

Area of circle = πr² = 22/7 × 56/11 × 56/11 = 896/11 = 81.45 cm²

∴ The area of the circle is larger than the area of the square

⇒ The area of the circle will be the largest.

∴ None of the given statements is required to answer the question.