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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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Video Explanation

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Explanatory Answer

Method of solving this CAT Question from Geometry

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.