Question

A rhombus has an area equal to half the sum of the areas of the squares built on its four sides. The ratio of the long diagonal to the short diagonal is:

(a) 2 + √3

(b) 2 – √3

(c) \(\frac 12\)

(d) Can't be determined

Answer 1

Correct option is (d) Can't be determined

Let the side of the rhombus be x.

Then, the area of the square on its side will be x²

And, the sum  of area of 4 squares on its side will be 4x²

So, the area of rhombus = 4x²/2 ​= 2x²

Let the diagonals be d₁​ and d_2​

Hence the area = 1/2​(d₁​×d₂​) = 2x²

⇒ d₁​ × d₂ ​= 4x²...(1)

And the diagonal of rhombus bisect each other at 90^∘, hence they form a right triangle.

So, by Pythagoras theorem

Hence, from (1) and (2), we cannot get a direct relation between the diagonals.