Problem:
Sides AB and AC of equilateral triangle ABC are tangent to a circle at points B and C, respectively. What fraction of the area of β³ABC lies outside the circle?
Answer Choices:
A. 2743βΟββ31β
B. 23βββ8Οβ
C. 21β
D. 3ββ923βΟβ
E. 34ββ2743βΟβ
Solution:
Let O be the center of the circle, and without loss of generality, assume that radius OB=1. Because β³ABO is a 30β60β90β right triangle, AO=2 and AB=BC=3β. Kite ABOC has diagonals of lengths 2 and 3β, so its area is 3β. Because β BOC=120β, the area of the sector cut off by β BOC is 31βΟ. The area of the portion of β³ABC lying outside the circle (shaded in the figure) is therefore 3ββ31βΟ. The area of β³ABC is 41β3β(3β)2=43β3β, so the requested fraction is
