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.