Question

if in two circles ,arc of the same length subtend angles 60°and 75°at the centre , find the ratio of their radii​

Answer 1

$\huge\color{navy} \mathtt{Answer}$

$\rm{we \: know \: that} \ \\ \boxed{l = r \theta}$

$\sf{there \: are \: 2 \: circles \: of \: different \: radius}$

$\sf{ \therefore \: the \: radius \: be \: denoted \: by \: r_1 \: and \: r_2}$

$\sf{length \: of \: arc \: of \: {l}^{st} \: circle}$

$l=r_1\theta\\ l= r_1 \times 60\degree \\ l= r_1 \times 60\degree \times \frac{\pi}{180 \degree } \\ l= r_1 \frac{\pi}{3}$

$\sf{length \: of \: arc \: of \: {ll}^{nd} \: cirlce }$

$l = r_2 \theta \\ l = r_2 \times 75 \degree \\ l = r_2 \times 75 \degree \times \frac{\pi}{180 \degree} \\ l = r_2 \times \frac{5\pi}{12}$

$\sf{it \: is \: given \: that \: arc \: are \: of \: same \: length \: }$

$\color{teal}{ \because \: length \: of \: {l}^{st} arc = length \: of \: {ll}^{nd} \: arc }$

$\large{r_1 \times \frac{\pi}{3} = r_2 \times \frac{5\pi}{12} }$

$\frac{r_1}{r_2 }= \frac{5\pi}{12} \times \frac{3}{\pi} \\ \frac{r_1}{r_2} = \frac{5}{4}$

$\therefore \: \sf{r_1 \ratio \: r_2 = 5 \ratio4}$