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Q.
If the arcs of the same lengths in two circles subtend angles $65^{\circ}$ and $110^{\circ}$ at the centre, then the ratio of their radii is equal to
Trigonometric Functions
Solution:
Let $r_1$ and $r_2$ be the radii of the two circles.
Given that, $\theta_1=65^{\circ}=\frac{\pi}{180} \times 65=\frac{13 \pi}{36}$ radian
and $ \theta_2=110^{\circ}=\frac{\pi}{180} \times 110=\frac{22 \pi}{36}$ 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$.