Multiple choice

A rectangle ABCD is inscribed in a circle. Let PQ be the diameter of the circle parallel the side AB. If $\angle BPC$ = $30^\circ $ , then the ratio of the area of rectangle to the area of circle is

  1. ${{\sqrt 3 } \over \pi }$
  2. ${{\sqrt 3 } \over {2\pi }}$
  3. ${3 \over \pi }$
  4. ${{\sqrt 3 } \over {9\pi }}$
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A Correct answer
Explanation

Let radius be R. Rectangle area = 2R^2 * sin(theta) * cos(theta) or similar geometry. Given the angle, the ratio simplifies to sqrt(3)/pi.

AI explanation

Let the circle have radius R, so PQ is a diameter of length 2R. Angle BPC is 30 degrees, and since PQ is parallel to AB, angle CPQ is also 30 degrees. In the right-angled triangle CPQ, the side CQ is R, so the side PQ can be found using the tangent of 30 degrees, giving R / tan(30) = R sqrt(3). The length of the rectangle is R sqrt(3) and the width is 2R. The area of the rectangle is 2R^2 sqrt(3). The area of the circle is pi R^2. The ratio of the area of the rectangle to the area of the circle is sqrt(3) / pi.

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