Question

If two arcs of the same length in two circles subtend angles 65° and 110° at the centre. Find the ratio of their radii.

Answer 1

Let r₁ and r₂ be the radii of the two circles and let their arcs of the same length S subtend angles of 65° and 110° at their centres.

Angle subtended at the centre of the first circle,

θ₁ = 65°

= `(65 xx pi/180)^"c"`

= `((13pi)/36)^"c"`

∴ S = r₁θ₁ = `"r"_1((13pi)/36)`  ...(i)

Angle subtended at the centre of the second circle,

θ₂ = 110°

= `(110 xx pi/180)^"c"`

= `((11pi)/18)^"c"`

∴ S = r₂θ₂ = `"r"_2((11pi)/18)`  ...(ii)

From (i) and (ii), we get

`"r"_1((13pi)/36) = "r"_2((11pi)/18)`

∴ `"r"_1/"r"_2 = 11/18 xx 36/13`

∴ `"r"_1/"r"_2 = 11/1 xx 2/13`

∴ `"r"_1/"r"_2 = 22/13`

∴ r₁ : r₂ = 22 : 13