Question

Can someone answer with steps and explanation? Thanks.

Answer 1

Answer:

$BC=9\sqrt{3}\approx15.59\text{ units}$

Step-by-step explanation:

Since Segment BOA is a diameter:

$m\angle ACB=90$

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:

$2x+4x=180$

Hence:

$x=30$

Thus, Arc CB = 120°. By the Inscribed Angle Theorem:

$\displaystyle m\angle A=\frac{1}{2}\left(\stackrel{\frown}{CB}\right)=\frac{1}{2}\left(120)=60$

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:

$BC=9\sqrt{3}\approx15.59\text{ units}$