Question

The square ABCD has a side length of 1, and AEF is an equilateral triangle. What is the area of AEF?

Answer 1

2sqrt(3)-3

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Area=sqrt(3)/4* a^2, where a= side of equilateral triangle

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BE=x --> EC=1-x --> CF=1-x, FD=x
From triangle ABE ---> a^2 = 1+x^2
From triangle ECF ---> a^2 = 2*(1-x)^2
1+x^2=2x^2-4x+2 -> x^2-4x+1=0 -> x=2-sqrt(3) --> a^2=x^2+1=8-4sqrt(3)
A=sqrt(3)/4*a^2=sqrt(3)/4*(8-4sqrt(3))=2sqrt(3)-3