Discussion

Explanation:

In □EFGH, EG is the diagonal. 

Also, EI and GJ are the perpendicular bisectors of the equilateral triangles AEB and GCD. 

Let us suppose AB = ’a’ units.

BC will also be ‘a’ units since AB and BC are sides of the same square.
 
In ∆ABE, EI = a√3/2 (perpendicular of an equilateral triangle is √3/2 times the side)

Similarly, GJ = a√3/2
 
Also, IJ = BC = a

∴ EG = EI + IJ + GJ = a√3/2 + a + a√3/2 = a(√3 + 1)

∴ Side of square EFGH = a(√3 + 1)/√2

⇒ Area of square EFGH = [a(√3 + 1)/√2]2 = (2 + √3)a2

Also, area of square ABCD = a2

Area of square ABCD = a2

Ratio of area of EFGH to area of ABCD = (2 + √3)a2 : a2 = (2 + √3) : 1

Hence, option (d).

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