Question

Two arcs of the same lengths subtend angles of 60° and 75° at the centres of two circles. What is the ratio of radii of two circles?

Answer 1

Let r₁ and r₂ be the radii of the two circles and let their arcs of same length S subtend angles of 60° and 75° at their centres.

Angle subtended at the centre of the first circle,

θ₁ = 60°

= `(60 xx pi/180)^"c"`

= `(pi/3)^"c"`

∴ S = r₁θ₁ = `"r"_1(pi/3)`  ...(i)

Angle subtended at the centre of the second circle,

θ₂ = 75°

= `(75 xx pi/180)^"c"`

= `((5pi)/12)^"c"`

∴ S = r₂θ₂ = `"r"_2((5pi)/12)`  ...(ii)

From (i) and (ii), we get

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

∴ `"r"_1/"r"_2 = 15/12`

∴ `"r"_1/"r"_2 = 5/4`

∴ r₁ : r₂ = 5 : 4.