The largest possible equilateral triangle is inscribed in a square. Triangle's vertices must touch either side of a square or its vertex. Find the ratio of the area of a triangle to the area of a square.
+14985 The largest possible equilateral triangle is inscribed in a square. Triangle's vertices must touch either side of a square or its vertex. Find the ratio of the area of a triangle to the area of a square.
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\(A_◽=a^2\\ A_{\ △}=\dfrac{s^2\sqrt{3}}{4}\)
\(cos(15°)=\dfrac{a}{s}\\ s^2=\dfrac{a^2}{cos^2(\dfrac{\pi}{12})}\)
\(cos(\dfrac{\pi}{12})=\dfrac{\sqrt{6}+\sqrt{2}}{4}\\ cos^2(\dfrac{\pi}{12})=\dfrac{8+2\sqrt{12}}{16}\)
\(cos^2(\dfrac{\pi}{12})=\dfrac{4+\sqrt{12}}{8}=\color{blue}\dfrac{4+2\sqrt{3}}{8}\)
\(s^2=\dfrac{a^2}{cos^2(\dfrac{\pi}{12})}\)
\(s^2=\dfrac{8a^2}{4+2\sqrt{3}}=\dfrac{4a^2}{2+\sqrt{3}}\)
\(A_{\ △}=\dfrac{s^2\sqrt{3}}{4}\)
\(A_{\ △}=\dfrac{4\cdot \sqrt{3}\cdot a^2}{4\cdot (2+\sqrt{3})}=\dfrac{\sqrt{3}}{2+\sqrt{3}}\cdot a^2\)
\(A_◻:A_△ =1:\dfrac{\sqrt{3}}{2+\sqrt{3}}\)
\(A_◻:A_△ =1:0,4641\)
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