Points E, F, G, H lie on the sides AB, BC, CD, and DA

Points E, F, G, H lie on the sides AB, BC, CD, and DA, respectively, of a square ABCD. If EFGH is also a square whose area is 62.5% of that of ABCD and CG is longer than EB, then the ratio of length of EB to that of CG is

  1. 2 : 5
  2. 4 : 9
  3. 3 : 8
  4. 1 : 3

Answer

Let the area of square ABCD be 100.

Side of ABCD = 10

Area of EFGH = 62.5

Side of EFGH = √62.5

Triangles AEH, BFE, CGF and DHG are congruent by ASA.

Let AE = BF = CG = DH = x

EB = FC = DG = AH = 10 - x

AE2 + AH2 = EH2

x2 + (10 - x)2 = (√62.5)2

Solving,

x = 2.5 or 7.5

Since, CG is longer than EB,

CG = 7.5 and EB = 2.5

Therefore, EB : CG = 1 : 3 

The correct option is D.