Question

If a chord $8$ inches long has an arc of ${120}^o$, the radius, in inches, of the circle is

• A. $4$
• B. $4\sqrt{3}$
• C. $\frac{8\sqrt{3}}{5}$
• D. $\frac{8\sqrt{3}}{3}$
• E. $2\sqrt{3}$

Answer 1

Answer: (D)

$\frac{8\sqrt{3}}{3}$

Given that the chord of length $8$ makes $120$ degrees angle at center of circle.
Let the radius of circle be $r$
The chord forms a triangle with center of circle
So, by sine rule, we get

$\frac{r}{\sin 30}=\frac{8}{\sin 120}$
$\Rightarrow x=8\frac{\sin 30}{\sin 60}$

$=8\times \frac{1/2}{\sqrt{3}/2}$

$=\frac{8\sqrt{3}}{3}$