Question

If the 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 the angles subtended at the centres by the arcs and radii of the first and second circles be ${\theta }_1\mathrm{and}{r}_1\mathrm{and}{\theta }_2\mathrm{and}{r}_2$, respectively.
Thus, we have:
${\theta }_1=65°=\left(65\times {\displaystyle \frac{\mathrm{\pi }}{180}}\right)\mathrm{radian}$
${\theta }_2=65°=\left(110\times {\displaystyle \frac{\mathrm{\pi }}{180}}\right)\mathrm{radian}$
${\mathrm{\theta }}_1=\frac{l}{r_1}$
$\Rightarrow r_1=\frac{l}{\left(65\times {\displaystyle \frac{\mathrm{\pi }}{180}}\right)}$

${\mathrm{\theta }}_2=\frac{l}{r_2}$
$\Rightarrow r_2=\frac{l}{\left(110\times {\displaystyle \frac{\mathrm{\pi }}{180}}\right)}$

$\Rightarrow \frac{r_1}{r_2}=\frac{\frac{l}{\left(65\times {\displaystyle \frac{\mathrm{\pi }}{180}}\right)}}{\frac{l}{\left(110\times {\displaystyle \frac{\mathrm{\pi }}{180}}\right)}}=\frac{110}{65}=\frac{22}{13}$

⇒ r₁:r₂ = 22:13