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A circle of maximum area is inscribed in the square. An equilateral triangle is inscribed in that circle and all the vertices of the triangle touch the circle. A triangle is formed by the joining of the midpoint of the sides of the equilateral triangle. Find the ratio of the area of the square and area of the triangle which is formed by midpoints.

A circle of maximum area is inscribed in the square. An equilateral triangle is inscribed in that circle and all the vertices of the triangle touch the circle. A triangle is formed by the joining of the midpoint of the sides of the equilateral triangle. Find the ratio of the area of the square and area of the triangle which is formed by midpoints. Correct Answer 64 : 3√3

Let the side of the square = a cm

Radius of the circle is inscribed in the square = a/2

Circumradius of an equilateral triangle = side of triangle/√3

⇒ Side of the triangle = (a/2) × √3 = (a√3/2)

Area of the triangle which is formed by the joining the midpoint of sides of the equilateral triangle = Area of an equilateral triangle/4

⇒ /4 = (3√3/64)a2

Area of the square : Area of the triangle formed in an equilateral triangle

⇒ a2 : (3√3/64) a2

⇒ 64 : 3√3

∴ Required Ratio is 64 : 3√3

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