Multiple choice

If arcs of same length in two circles subtend angles of $60^{\circ}$ and $75^{\circ}$ at their center, find the ratios of their radii.

  1. $r_{1}:r_{2}=5:4$
  2. $r_{1}:r_{2}=4:5$
  3. $r_{1}:r_{2}=5:3$
  4. $r_{1}:r_{2}=3:5$
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A Correct answer
Explanation

Since the arc lengths are equal, the product of the radius and the angle (in radians) must be equal for both circles: r1 * theta1 = r2 * theta2. Thus, r1/r2 = theta2/theta1 = 75/60 = 5/4.

AI explanation

If two arcs have the same length l, then l = (theta1 / 360) * 2 * pi * r1 and l = (theta2 / 360) * 2 * pi * r2. Equating the two expressions gives (theta1) * r1 = (theta2) * r2. Substituting the given angles 60 and 75 results in 60 * r1 = 75 * r2, which simplifies to r1 / r2 = 75 / 60 = 5 / 4. Therefore, the ratio of their radii is r1:r2 = 5:4.

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