Problem:
In a circle with center O,AD is a diameter, ABC is a chord, BO=5 and ∠ABO=CD=60∘. Then the length of BC is
Answer Choices:
A. 3
B. 3+3​
C. 5−23​​
D. 5
E. none of the above
Solution:
Consider the half circle ACDO. Let 2θ=60∘=∠ABO=CD. Then ∠CAD=θ, since it is an inscribed angle. Draw in CO. Since △COA is isosceles, ∠ACO=θ. By the Exterior Angle Theorem applied to △BOC at B,∠BOC=2θ−θ=θ. So △BOC is also isosceles and BC=5.
Note. The fact that 2θ=60∘ is irrelevant. BC=5 for any θ for which the configuration exists. What is the range of values for θ?
