2014 AMC 12A Problems/Problem 12
Problem
Two circles intersect at points and . The minor arcs measure on one circle and on the other circle. What is the ratio of the area of the larger circle to the area of the smaller circle?
Solution
Let the radius of the larger and smaller circles be and , respectively. Also, let their centers be and , respectively. Then the ratio we need to find is Draw the radii from the centers of the circles to and . We can easily conclude that the belongs to the larger circle, and the degree arc belongs to the smaller circle. Therefore, and . Note that is equilateral, so when chord AB is drawn, it has length . Now, applying the Law of Cosines on : (Solution by brandbest1)