Multiple choice

A chord of length 5 cm subtends an angle of 60° at the centre of a circle. The length (in cm) of a chord that subtends an angle of 120° at the centre of the same circle is

  1. 5 3

  2. 6 2

  3. 2 π

  4. 8
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A Correct answer
Explanation

A chord subtending central angle θ has length 2R sin(θ/2). The 60° chord gives 5 = 2R sin 30°, so R = 5, and the 120° chord is 2 x 5 x sin 60° = 5sqrt(3) cm.

AI explanation

A 60 degree central angle forms an isosceles triangle with two radii and the 5 cm chord, meaning the triangle is equilateral and the radius is 5 cm. For a 120 degree central angle, the perpendicular from the center bisects both the angle and the new chord, creating a right triangle with a hypotenuse of 5 and an angle of 60 degrees. Using the sine ratio, half the chord length is 5 * sin(60) = 5 * sqrt(3)/2, so the full chord length is 5 * sqrt(3) cm.

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