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 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

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

Thus, the ratio of the radii is 5 : 4.