A circle is inscribed in an equilateral triangle of side a. What is the area of any square inscribed in this circle? -Maths 9th

A circle is inscribed in an equilateral triangle of side a. What is the area of any square inscribed in this circle? -Maths 9th
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1 Answer

Answer :

(c) \(rac{a^2}{6}.\)If ‘a’ is length of the side of ΔABC, thenArea of ΔABC = \(rac{\sqrt3}{4}\,a^2\)semi-perimeter of ΔABC = \(rac{3a}{2}\)∴ Radius of in-circle = \(rac{ ext{Area}}{ ext{semi-perimeter}}\) = \(rac{\sqrt3}{4}\,a^2\) x \(rac{2}{3a}\) = \(rac{a}{2\sqrt3}\)∴ Diagonal of square PQRS = Diameter of incircle.= 2 x \(rac{a}{2\sqrt3}\) = \(rac{a}{\sqrt3}\) ∴ Area of squre = \(rac{( ext{diagonal})^2}{2}\) = \(rac{\big(rac{a}{\sqrt3}\big)^2}{2}\) = \(rac{a^2}{6}.\)
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