If the arcs of the same lengths in two circles subtend angles $65^{\circ}$ and $110^{\circ}$ at the centre, find the ratio of their radii.
If the arcs of the same lengths in two circles subtend angles $65^{\circ}$ and $110^{\circ}$ at the centre, find the ratio of their radii.
Let $r_{1}$ and $r_{2}$ be the radii of the two circles. Given that
${{\theta _1} = {{65}^\circ } = \frac{\pi }{{180}} \times 65 = \frac{{13\pi }}{{36}}\,{\text{ radian }}}$
and ${{\theta _2} = {{110}^\circ } = \frac{\pi }{{180}} \times 110 = \frac{{22\pi }}{{36}}{\text{ }}\,{\text{radian }}}$
Let $l$ be the length of each of the arc. Then $l=r_{1} \theta_{1}=r_{2} \theta_{2},$ which gives
$\frac{13 \pi}{36} \times r_{1}=\frac{22 \pi}{36} \times r_{2}, \text { i.e., } \frac{r_{1}}{r_{2}}=\frac{22}{13}$
Hence $r_{1}: r_{2}=22: 13$
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