Multiple choice

Two parallel chords are drawn on the same side of the centre of a circle of radius 20. It is found that they subtend ${ 60 }^{ 0 }$ and ${ 120 }^{ 0 }$ angles at the centre of the circle. Then the perpendicular distance between the chords is:

  1. $5(\sqrt { 3 } -1)$
  2. $10(\sqrt { 3 } -1)$
  3. $10(\sqrt { 2 } -1)$
  4. $5(\sqrt { 3 } +1)$
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B Correct answer
Explanation

The distance from the center to a chord is d = R * cos(theta/2). For the 60 degree angle, the distance is 20 * cos(30) = 20 * (sqrt(3)/2) = 10 * sqrt(3). For the 120 degree angle, the distance is 20 * cos(60) = 20 * (1/2) = 10. Since both chords are on the same side, the perpendicular distance between them is 10 * sqrt(3) - 10 = 10 * (sqrt(3) - 1).

AI explanation

The perpendicular distance from the center of a circle to a chord is given by the formula d = r cos(theta/2). For the chord subtending 120 degrees, the distance from the center is 20 cos(60 degrees) = 20 * (1/2) = 10. For the chord subtending 60 degrees, the distance is 20 cos(30 degrees) = 20 * (sqrt(3)/2) = 10 sqrt(3). The distance between the two parallel chords on the same side of the center is the difference of these distances: 10 sqrt(3) - 10 = 10(sqrt(3) - 1). The result is 10(sqrt(3) - 1).

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