Multiple choice

In a circle with centre O, AB is a chord which subtends an angle of 30° at the centre. E and F are mid-points of OA and OB, respectively. Which of the following gives the ratio of the length of EF to that of the minor arc AB?

  1. 4√(2√3-1) / π

  2. 3√(2-√3) / π

  3. 2√(5-2√3) / π

  4. None of the above
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B Correct answer
Explanation

Using geometry of the circle and midpoints, the length of EF is related to the chord AB. The ratio calculation involves the chord length and the arc length formula.

AI explanation

Taking the radius of the circle as R, the chord AB is given by 2R sin(15 degrees) using the half-angle formula for the 30 degree central angle. Since E and F are midpoints of OA and OB, triangle OEF is similar to triangle OAB, making EF exactly half the length of AB, which equals R sin(15 degrees). The length of the minor arc AB is calculated using the arc length formula R theta, yielding R times pi over 6. The ratio of EF to the minor arc is R sin(15 degrees) divided by R pi over 6, and replacing sin(15 degrees) with its exact value of the square root of (2 minus the square root of 3) divided by 2 simplifies the ratio to 3 times the square root of (2 minus the square root of 3) divided by pi.

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