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XAT 2019 — QA & DI Question 13

Basics of QuadrilateralsMedium

In the picture below, EFGH, ABCD are squares, and ABE, BCF, CDG, DAH are equilateral triangles. What is the ratio of the area of the square EFGH to that of ABCD? 

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Answer & solution

  • A

    √2 + 2

  • B

    3 + √2

  • C

    √2 + √3

  • √3 + √2

  • E

    1 + √3

Solution

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).

XAT 2019 QA & DI Q13: In the picture below, EFGH, ABCD are squares, and ABE, BCF, CDG, DAH are equilateral triangles. What is the ra — Solution | TheCATExam