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

Let the radii of the two circles be r1 and r2. Let an arc of length l subtend an angle of 60° at the centre of the circle of radius r1, while let an arc of length l subtend an angle of 75° at the centre of the circle of radius r2.

Now, 60° = π/3 radian and 75° = 5π/12 radian

We know that in a circle of radius r unit, if an arc of length l unit subtends an angle θ radian at the centre, then θ = l/r or l = rθ

∴ l = r1(π/3) and l = r2(5π/12)

⇒ r1(π/3) = r2(5π/12)

⇒ r1 = r2(5/4)

⇒ r1/r2 = 5/4