# If the arcs of the same length in two circles subtend angles of $60^{\large\circ}$ and $75^{\large\circ}$ at their respective centres,find the ratio of their radii

$\begin{array}{1 1}(A)\;2 : 3&(B)\;3 : 2\\(C)\;4 : 5&(D)\;5 : 4\end{array}$

Toolbox:
• Radian measure =$\large\frac{\pi}{180}$$\times Degree measure • l=r\theta Let r_1 and r_2 be the radii of the two circles,then \theta_1=60^{\large\circ} \Rightarrow (60\times \large\frac{\pi}{180})^c=(\frac{\pi}{3})^c \theta_2=75^{\large\circ} \Rightarrow (75\times \large\frac{\pi}{180})^c=(\frac{5\pi}{12})^c Let the length of each arc be 'l' cm,then l=r_1\theta_1=r_2\theta_2 \Rightarrow (r_1\times \large\frac{\pi}{3})$$=(r_2\times \large\frac{5\pi}{12})$
$\Rightarrow \large\frac{r_1}{r_2}=\frac{5\pi}{12}\times \frac{3}{\pi}$
$\Rightarrow \large\frac{r_1}{r_2}=\frac{5}{4}$
Hence $r_1 : r_2=5 : 4$
Hence (D) is the correct answer.