Q15 from Senior Mathematical Challenge 2018: A square is inscribed in a circle of radius 1. An isosceles triangle is inscribed in the square. What is the ratio of the area of this triangle to the area of the shaded region? (Requires Diagram))
Radius = 1, therefore diameter = 2Let x be the length of one side of the square.Using Pythagoras,x2 + x2 = 222x2 = 4x = sqrt(2)Area of isosceles triangle = side of square * half side of square= sqrt(2) * sqrt(2)/2= 1Shaded area = area of circle - area of square= π(1)2 - sqrt(2)2 = π - 2 Answer = 1:π-2
