Problem:
Two circles intersect at points A and B. The minor arcs AB measure 30∘ on one circle and 60∘ on the other circle. What is the ratio of the area of the larger circle to the area of the smaller circle?
Answer Choices:
A. 2
B. 1+3
C. 3
D. 2+3
E. 4 Solution:
Let the larger and smaller circles have radii R and r, respectively. Then the length of chord AB can be expressed as both r and 2Rsin15∘. The ratio of the areas of the circles is
πr2πR2=4sin215∘1=2(1−cos30∘)1=2−31=2+3
OR
Let the larger and smaller circles have radii R and r, and centers P and Q, respectively. Because △QAB is equilateral, it follows that r=AB. The Law
of Cosines applied to △PBA gives
