Question

If in two circles, arcs of the same length subtend angles 60° and 75° at the centre, find the ratio of their radii.

Answer 1

Let the radii of the two circles be r1 and r2 . Let an arc of length l subtend an angle of 60° at the centre of the circle of radiusr1, while let an arc of length l subtend an angle of 75° at the centre of the circle of radius r2

$Now, \,60° = \frac{\pi}{3} \text{radian and\,75°} =\frac{\pi}{12}\,radian$

We know that in a circle of radius r unit, if an arc of length l unit subtends an angle θ radian at the centre, then

$θ=\frac{l}{r}\,or\,l=rθ$

$l=\frac{r_1{\pi}}{3}\,and\,l=\frac{r_25{\pi}}{12}$

$=\frac{r_1{\pi}}{3}\,and\,l=\frac{r_25{\pi}}{12}$

$r_1=\frac{r_25}{4}$

$\frac{r_1}{r_2}=\frac{5}{4}$

Thus, the ratio of the radii is 5:4.