Answer:
[tex]BC=9\sqrt{3}\approx15.59\text{ units}[/tex]
Step-by-step explanation:
Since Segment BOA is a diameter:
[tex]m\angle ACB=90[/tex]
Arc Ac and Arc CB are in a ratio of two to four. Since Segment BOA is a diameter, Arc ACB measures 180°. Letting the unknown value be x, we can write that:
[tex]2x+4x=180[/tex]
Hence:
[tex]x=30[/tex]
Thus, Arc CB = 120°. By the Inscribed Angle Theorem:
[tex]\displaystyle m\angle A=\frac{1}{2}\left(\stackrel{\frown}{CB}\right)=\frac{1}{2}\left(120)=60[/tex]
Therefore, ΔABC is a 30-60-90 triangle. Its sides are in the ratios shown in the image below.
Since AC is opposite from the 30° triangle, let AC = a.
We are given that AC = 9. Hence, a = 9.
BC is opposite from the 60° angle and it is given by a√3. Therefore:
[tex]BC=9\sqrt{3}\approx15.59\text{ units}[/tex]
