Find the area of an equilateral triangle inscribed in a circle circumscribed by a square made by joining the mid-points -Maths 9th

Find the area of an equilateral triangle inscribed in a circle circumscribed by a square made by joining the mid-points -Maths 9th
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1 Answer

Answer :

(d) \(rac{3\sqrt3a^2}{32}\)Let AB = a be the side of the outermost square.Then AG = AH = \(rac{a}{2}\)⇒ GH = \(\sqrt{rac{a^2}{4}+rac{a^2}{4}}\) = \(rac{a}{\sqrt2}\)∴ Diameter of circle = \(rac{a}{\sqrt2}\)⇒ Radius of circle = \(rac{a}{2\sqrt2}\)If O is the centre of the circle, then ∠POQ = 120º.∴ Area of ΔPOQ = \(rac{1}{2}\) x PO x PQ x sin 120º= \(rac{1}{2}\) x \(rac{a}{2\sqrt2}\) x \(rac{a}{2\sqrt2}\) x \(rac{\sqrt3}{2}\) = \(rac{\sqrt3a^2}{32}\)∴ Area of ΔPQR = 3 (Area of ΔPOQ) = \(rac{\sqrt3a^2}{32}\)
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