An equilateral

An equilateral triangle is inscribed in a square of side 1 unit long so that it has one common vertex with the square. What is the area of the inscribed triangle?

Correct result:

S =  0.464

Solution:

a2=12+x2 a2=(1x)2+(1x)2  1+x2=(1x)2+(1x)2  12+x2=(1x)2+(1x)2  x2+4x1=0 x24x+1=0  a=1;b=4;c=1 D=b24ac=42411=12 D>0  x1,2=b±D2a=4±122=4±232 x1,2=2±1.7320508075689 x1=3.7320508075689 x2=0.26794919243112   Factored form of the equation:  (x3.7320508075689)(x0.26794919243112)=0  0<x<1 x=x2=0.26790.2679  a=1+x2=1+0.267921.0353  S=34 a2=34 1.03532=0.464a^2=1^2+x^2 \ \\ a^2=(1-x)^2+(1-x)^2 \ \\ \ \\ 1+x^2=(1-x)^2+(1-x)^2 \ \\ \ \\ 1^2+x^2=(1-x)^2+(1-x)^2 \ \\ \ \\ -x^2 +4x -1=0 \ \\ x^2 -4x +1=0 \ \\ \ \\ a=1; b=-4; c=1 \ \\ D=b^2 - 4ac=4^2 - 4\cdot 1 \cdot 1=12 \ \\ D>0 \ \\ \ \\ x_{1,2}=\dfrac{ -b \pm \sqrt{ D } }{ 2a }=\dfrac{ 4 \pm \sqrt{ 12 } }{ 2 }=\dfrac{ 4 \pm 2 \sqrt{ 3 } }{ 2 } \ \\ x_{1,2}=2 \pm 1.7320508075689 \ \\ x_{1}=3.7320508075689 \ \\ x_{2}=0.26794919243112 \ \\ \ \\ \text{ Factored form of the equation: } \ \\ (x -3.7320508075689) (x -0.26794919243112)=0 \ \\ \ \\ 0<x<1 \ \\ x=x_{2}=0.2679 \doteq 0.2679 \ \\ \ \\ a=\sqrt{ 1+x^2 }=\sqrt{ 1+0.2679^2 } \doteq 1.0353 \ \\ \ \\ S=\dfrac{ \sqrt{ 3 } }{ 4 } \cdot \ a^2=\dfrac{ \sqrt{ 3 } }{ 4 } \cdot \ 1.0353^2=0.464

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