1. The minimum perimeter of the quadrilateral is

Question

The area of any cyclic quadrilateral ABCD is given by A² = (s – a)(s – b)(s – c)(s – d), where 2s =a + b + c + d, a, b, c and d are the sides of the quadrilateral. For a cyclic quadrilateral ABCD of area 1 sq. unit answer the following questions:

1. The minimum perimeter of the quadrilateral is

(A)  4 

(B)  2 

(C)  1 

(D)  None of these

2. The minimum value of the sum of the lengths of diagonals is

(A)  2√2

(B)  2

(C)  √2

(D) None of these

3.  When the perimeter is minimum the quadrilateral is necessarily 

(A) a square

(B) a rectangle but not a square

(C) a rhombus but not a square

(D) None of these

Answer 1

Correct option (A) (A)(A)

Applying AM ≥ GM for s – a, s – b, s – c, s – d

We have

2.   Also,

 

3.  When the perimeter is minimum,

s – a = s – b = s – c = s – d ⇒ a = b = c = d

So ABCD is a square.