Question

If in two circles arcs of same length subtend angles of ${60}^0$ and ${75}^0$ at the centre, then the ratio of their radii is:

• A. $4:5$
• B. $5:4$
• C. not derivable
• D. none of these

Answer 1

The correct option is A $4:5$

Arc length are same we need $\frac{r_2}{r_1}$

from fig $\left(1\right)$ :- $x=\left(2\pi R_1\right)\times \left(\frac{360}{75}\right)$

$R_1=\left(\frac{75}{360}\right)\times \frac{x}{2\pi }$

from fig $\left(2\right)$ :- $x=\left(2\pi r_2\right)\left(\frac{360}{60}\right)$

$R_2=\left(\frac{60}{360}\right)\left(\frac{x}{2\pi }\right)$

$\therefore \frac{R_2}{R_1}=\frac{\left(\frac{60}{360}\right)}{\left(\frac{75}{630}\right)}=\frac{60}{75}=\frac{4}{5}$

$\therefore R_2:R_1=4:5$