Question

If in two circles, arcs of the same length subtend angles 60° and 75° at the centre, find the ratio of their radii

Answer 1

Let r₁ and r₂ be the radii of the two circles and l be the length of the arc.

Given central angle θ₁ = 60°

θ₁ = `60 xx pi/180 = pi/3` radians

θ₂ = 75°

θ₂ = `75 xx pi/180 = (5pi)/12` radians

Given that l = r₁θ₁ = r₂θ₂ 

`"r"_1 pi/3 = "r"_2 (5pi)/12`

`"r"_1/"r"_2 = (5pi)/12 xx 3/pi`

`"r"_1/"r"_2 = 5/4`

r₁ : r₂ = 5 : 4