Inscribed Square Side Ratio

CAT 2019 Slot 1 · QA · Medium · Geometry

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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. A.

    1 : 3

  2. B.

    4 : 9

  3. C.

    2 : 5

  4. D.

    3 : 8

Answer

A

Explanation

Let side of ABCD be 1. Let EB=xEB = x, then AE=1−xAE = 1 - x. By symmetry of square EFGH, DG=EB=xDG = EB = x, CG=1−xCG = 1 - x. Side length squared of EFGH = EB2+BF2=x2+(1−x)2EB^2 + BF^2 = x^2 + (1 - x)^2. Area of EFGH = x2+(1−x)2=0.625=5/8x^2 + (1 - x)^2 = 0.625 = 5/8. 2x2−2x+1=5/8⇒16x2−16x+3=02x^2 - 2x + 1 = 5/8 \Rightarrow 16x^2 - 16x + 3 = 0. (4x−1)(4x−3)=0⇒x=1/4(4x - 1)(4x - 3) = 0 \Rightarrow x = 1/4 or x=3/4x = 3/4. Since CG>EB⇒1−x>x⇒x<1/2CG > EB \Rightarrow 1 - x > x \Rightarrow x < 1/2, so x=1/4x = 1/4. Then EB=1/4EB = 1/4 and CG=3/4CG = 3/4. Ratio EB:CG=1/4:3/4=1:3EB : CG = 1/4 : 3/4 = 1 : 3.

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