Question

If the length of an arc of a circle of radius r is equal to that of an arc of a circle of radius 2r, then the angle of the corresponding sector of the first circle is double the angle of the corresponding sector of the other circle. Is this statement false? Why?

Answer 1

Let two circles C₁ and C₂ of radius r and 2r with centres O and O’, respectively.

It is given that, the arc length `hat("AB")` of C₁ is equal to arc length `hat("CD")` of C₂

i.e., `hat("AB") = hat("CD") = l`  ...(say)

Now, let θ₁ be the angle subtended by arc `hat("AB")` and θ₂ be the angle subtended by arc `hat("CD")` at the centre.

∴ `hat("AB") = l = θ_1/360^circ xx 2π"r"`   ...(i)

And `hat("CD") = l = θ_2/360^circ xx 2π (2"r")`

= `θ_2/360^circ xx 4π"r"`  ...(ii)

From (i) and (ii), we get

`θ_1/360^circ xx 2π"r" = θ_2/360^circ xx 4π"r"`

⇒ θ₁ = 2θ₂

 i.e., Angle of the corresponding sector of C₁ is double the angle of the corresponding sector of C₂.