Multiple choice

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

  1. $4$
  2. $4\sqrt{3}$
  3. $\dfrac{8\sqrt{3}}{5}$
  4. $\dfrac{8\sqrt{3}}{3}$
  5. $2\sqrt{3}$
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D Correct answer
Explanation

In a circle, a chord of length L subtending an angle theta at the center has radius R = L / (2 * sin(theta/2)). Here, L = 8, theta = 120, so theta/2 = 60. R = 8 / (2 * sin(60)) = 8 / (2 * sqrt(3)/2) = 8 / sqrt(3) = 8 * sqrt(3) / 3.

AI explanation

Using the chord length formula $l = 2r \sin(\theta/2)$, we substitute $l = 8$ and $\theta = 120^\circ$ to get $8 = 2r \sin(60^\circ)$. Solving for the radius yields $8 = 2r(\sqrt{3}/2)$, which simplifies to $8 = r\sqrt{3}$. Rationalizing the denominator gives the radius as $8\sqrt{3}/3$.

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