A square is inscribed in a circle, and this circle is inscribed in an equilateral triangle. If the area of the square is 72 sq. units and the area of the equilateral triangle is k 3 sq. units, then the value of k is

Step-by-step solution

Let the side of the square be x . Then x^2 = 72 , which gives x = 6 2 . The diagonal of the square is 2 x = 12 . Since the square is inscribed in the circle, the diameter of the circle is equal to the diagonal of the square. Thus, the radius of the circle is r = 12 2 = 6 . This circle is inscribed in an equilateral triangle of side a . The inradius of an equilateral triangle is given by r = a 2 3 . 6 = a 2 3 a = 12 3 . The area of the equilateral triangle is 3 4 a^2 = 3 4 (12 3 )^2 = 3 4 (432) = 108 3 . Comparing t