Question

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

• A. $r_1:r_2=5:4$
• B. $r_1:r_2=4:5$
• C. $r_1:r_2=5:3$
• D. $r_1:r_2=3:5$

Answer 1

The correct option is A $r_1:r_2=5:4$

Let $r_1$ and $r_2$ be the radii of the given circles and let their arcs be of same length s subtending angles of ${60}^{\circ }$ and ${75}^{\circ }$ at their centers. Now,
${60}^{\circ }$ = $(60\times \frac{\pi }{180}{)}^R$ = ${\left(\frac{\pi }{3}\right)}^R$ and ${75}^0={(75\times \frac{\pi }{180})}^R={\left(\frac{5\pi }{12}\right)}^R$
$\therefore \frac{\pi }{3}=\frac{s}{r_1}$ and $\frac{5\pi }{12}=\frac{s}{r_2}$
$\Rightarrow \frac{\pi }{3}{r}_1=s$ and $\frac{5\pi }{12}{r}_2=s$
$\Rightarrow \frac{\pi }{3}{r}_1=\frac{5\pi }{12}{r}_2$
$\Rightarrow 4r_1=5r_2$
$\Rightarrow r_1:r_2=5:4$