Question

If in two circles, arcs of the same length subtend angles ${60}^{\circ }$ and ${75}^{\circ }$ at the centre, find the ratio of their radii.

Answer 1

Let the radii of the two circles be $r_1$ and $r_2$. Let an arc of length I subtend an angle of ${60}^{\circ }$ at the centre of the circle of radius $r_1$, while let an arc of length I subtend an angle of ${75}^{\circ }$ at the centre of the circle of radius $r_2$.
Now, ${60}^{\circ }=\frac{\pi }{3}$ radian and ${75}^{\circ }=\frac{5\pi }{12}$ radian
We know that in a circle of radius r unit, if an arc of length l unit subtends an angle $\theta$ radian at the centre, then
$\theta =\frac{l}{r}$ or $l=r\theta$
$\therefore l=\frac{r_1\pi }{3}$ and $l=\frac{r_{2}5\pi }{12}$
$⟹\frac{r_1\pi }{3}=\frac{r_{2}5\pi }{12}$
$⟹r_1=\frac{r_{2}5}{4}$
$⟹\frac{r_1}{r_2}=\frac{5}{4}$