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Nested Squares and Equilateral Triangles

XAT 2019 · Quantitative Ability & Data Interpretation · Hard · Geometry

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In the picture below, EFGH, ABCD are squares, and ABE, BCF, CDG, DAH are equilateral triangles. What is the ratio of the area of the square EFGH to that of ABCD?

  1. A.

    3+2\sqrt{3} + 2

  2. B.

    2+3\sqrt{2} + \sqrt{3}

  3. C.

    1+31 + \sqrt{3}

  4. D.

    2+2\sqrt{2} + 2

  5. E.

    3+23 + \sqrt{2}

Answer

A

Explanation

Let side of square ABCD be xx. Height of equilateral triangle with side xx is 32x\frac{\sqrt{3}}{2}x. Length of diagonal EFGH=x+2(32x)=x(1+3)EFGH = x + 2 \left(\frac{\sqrt{3}}{2}x\right) = x(1 + \sqrt{3}). Side of square EFGH y=diagonal2=x1+32y = \frac{diagonal}{\sqrt{2}} = x \frac{1+\sqrt{3}}{\sqrt{2}}. Ratio of areas =y2x2=(1+32)2=1+3+232=2+3= \frac{y^2}{x^2} = \left(\frac{1+\sqrt{3}}{\sqrt{2}}\right)^2 = \frac{1 + 3 + 2\sqrt{3}}{2} = 2 + \sqrt{3}.

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