Multiple choice

Area of circle in which a chord of length $ 2 \sqrt3\ units$, subtends angle $ 120^o$ at its centre is :

  1. $ \pi \ sq\ units$
  2. $ 2 \pi \ sq\ units$
  3. $ 4 \pi \ sq\ units$
  4. $ 5 \pi \ sq\ units$
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C Correct answer
Explanation

In a circle, a chord of length L subtends angle theta at the center. L = 2R sin(theta/2). 2*sqrt(3) = 2R sin(60). sqrt(3) = R * (sqrt(3)/2), so R = 2. Area = pi * R^2 = pi * 2^2 = 4pi.

AI explanation

Let the radius be r. The perpendicular from the center to the chord bisects the chord and the central angle, creating a right triangle with a 60 degree angle, a half-chord of the square root of 3, and the hypotenuse r. Using the sine ratio, r equals the square root of 3 divided by sin 60 degrees, which yields r equals 2. The area of the circle is pi times r squared, so the area is 4 pi square units.

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