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Question 24 :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. 1 : 3
  2. 4 : 9
  3. 2 : 5
  4. 3 : 8

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Explanatory Answer

Method of solving this CAT Question from Geometry
Geometry - Finding the ratio EB : CG

Given, Area(EFGH) = 62.5% Area(ABCD)
Area(EFGH) = \\frac{5}{8})Area(ABCD)
Let EB = 1 and CG = r
Similarly, the rest other dimensions are also of lengths 1 or r units
Applying pythagoras theorem on △DHG we get GH = √(1+r2)
Area(ABCD) = (1+r)2 and Area(EFGH) = (√(1+r2))2
=> (1+r2) = \\frac{5}{8})(1+r)2

1 + r2 = \\frac{5}{8})1+r2+2r)
8 + 8r2 = 5 + 5r2 + 10r
3r2 – 10r + 3 = 0
(3r-1)×(r-3) = 0
r = \\frac{1}{3}) or r = 3
As CG > EB
r = 3 and ratio (EB : CG)= (1:r) = (1:3)

The question is "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:"

Hence, the answer is 1 : 3

Choice A is the correct answer.

 

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