 # If the length of an arc of a circle of radius r

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?

Solution:

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

It is given that, the arc length $\mathrm{AB}$ of $C_{1}$ is equal to arc length $C D$ of $C_{2} i, e, A B=C D=l$ (say) Now, let $\theta$, be the angle subtended by arc $\overparen{A B}$ of $\theta_{2}$ be the angle subtended by arc $\overparen{C D}$ at the

$\therefore \quad \widehat{A B}=l=\frac{Q_{1}}{360} \times 2 \pi r$ $\ldots$ (i)

and $\overparen{C D}=l=\frac{\theta_{2}}{360} \times 2 \pi(2 r)=\frac{\theta_{2}}{360} \times 4 \pi r$...(ii)

From Eqs. (i) and (ii),

$\frac{\theta_{1}}{360} \times 2 \pi r=\frac{\theta_{2}}{360} \times 4 \pi r$

$\Rightarrow \quad \theta_{1}=2 \theta_{2}$

i.e., angle of the corresponding sector of C1 is double the angle of the corresponding sector of C2.
It is true statement