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 \[\theta_1\text{ and }r_1\text{ and }\theta_2\text{ and }r_2\] respectively.
Thus, we have:
\[\theta_1 = 65^\circ = \left( 65 \times \frac{\pi}{180} \right)\text{ radian }\]
\[\theta_2 = 65^\circ = \left( 110 \times \frac{\pi}{180} \right)\text{ radian }\]
\[\theta_1 = \frac{l}{r_1}\]
\[\Rightarrow r_1 = \frac{l}{\left( 65 \times \frac{\pi}{180} \right)}\]
\[\theta_2 = \frac{l}{r_2}\]
\[\Rightarrow r_2 = \frac{l}{\left( 110 \times \frac{\pi}{180} \right)}\]
\[\Rightarrow \frac{r_1}{r_2} = \frac{\frac{l}{\left( 65 \times \frac{\pi}{180} \right)}}{\frac{l}{\left( 110 \times \frac{\pi}{180} \right)}} = \frac{110}{65} = \frac{22}{13}\]

⇒ r₁:r₂ = 22:13