# CAT 2018 Question Paper | Quants Slot 1

###### CAT Previous Year Paper | CAT Geometry Questions | Question 24

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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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##### 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.

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