Multiple choice

Two equal arcs of different circles C1 and C2 subtend angles of 60° and 75° respectively, at the centres. What is the ratio of the radius of C1 to the radius of C2?

  1. 4 : 5

  2. 5 : 4

  3. 1 : 1

  4. 3 : 2
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B Correct answer
Explanation

Arc length s = r * theta (in radians). s1 = r1 * 60 * (pi/180) = r1 * pi/3. s2 = r2 * 75 * (pi/180) = r2 * 5pi/12. Since s1 = s2, r1 * pi/3 = r2 * 5pi/12. r1/r2 = (5/12) / (1/3) = 5/4.

AI explanation

By the arc length formula, length equals the angle in radians times the radius. Since the arc lengths are equal, 60 degrees times r1 equals 75 degrees times r2. Converting the angles to a ratio gives r1 divided by r2 equals 75 divided by 60. This simplifies to a ratio of 5 : 4.

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