Question

If in two circle, arcs of the same length subtend angle ${60}^o$ and ${75}^o$ at the centre , find the ratio of their radii.

Answer 1

We know $l=r\theta$
Length of arc of $1^{st}$ circle
$l=r_1\theta =r_1\times 60°=r_1\times 60\times \frac{\pi }{180}=r_1\frac{\pi }{3}$
$2^{nd}$ circle
$l=r_2\theta =r_2\times 75°=r_2\times 75\times \frac{\pi }{180}=r_2\frac{5\pi }{12}$
As the arcs are of same length.
Length of $1st$ arc$=$length of $2nd$ arc
$r_1.\frac{\pi }{3}=r_1.\frac{\pi }{3}{r}_1:r_2=5:4$