If the arcs of the same length in two circles subtend angles 65° and 110° at the centre,
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.

Solution:

Let the angles subtended at the centres by the arcs and radii of the first and second circles be $\theta_{1}$ and $r_{1}$ and $\theta_{2}$ and $r_{2} \quad \theta 1$ and $r 1$ and $\theta 2$ and $r 2$, respectively.

Thus, we have:

$\theta_{1}=65^{\circ}=\left(65 \times \frac{\pi}{180}\right) \operatorname{radian}$

$\theta_{2}=65^{\circ}=\left(110 \times \frac{\pi}{180}\right) \operatorname{radian}$

$\theta_{1}=\frac{l}{r_{1}}$

$\Rightarrow r_{1}=\frac{l}{\left(65 \times \frac{\pi}{180}\right)}$

$\theta_{2}=\frac{l}{r_{2}}$

$\Rightarrow r_{2}=\frac{l}{\left(110 \times \frac{\pi}{180}\right)}$

$\Rightarrow \frac{r_{1}}{r_{2}}=\frac{\frac{l}{\left(65 \times \frac{\pi}{180}\right)}}{\frac{l}{\left(110 \times \frac{\pi}{180}\right)}}=\frac{110}{65}=\frac{22}{13}$

$\Rightarrow r_{1}: r_{2}=22: 13$