If the arcs of the same length in two circles | vedclass

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Question

(A) The formula for the angle subtended by an arc at the center is $	heta = \frac{l}{r}$,where $	heta$ is the angle in radians,$l$ is the arc length

Vedclass · Accepted Answer

$8:5$

Vedclass · Answer

(A) The formula for the angle subtended by an arc at the center is $	heta = \frac{l}{r}$,where $	heta$ is the angle in radians,$l$ is the arc length,and $r$ is the radius.Given that the arc lengths are equal,let $l_1 = l_2 = l$.We have $	heta_1 = 75^{\circ}$ and $	heta_2 = 120^{\circ}$.Converting degrees to radians: $	heta_1 = 75 	imes \frac{\pi}{180} = \frac{5\pi}{12}$ radians and $	heta_2 = 120 	imes \frac{\pi}{180} = \frac{2\pi}{3}$ radians.Since $r = \frac{l}{	heta}$,the ratio of the radii is $\frac{r_1}{r_2} = \frac{l/	heta_1}{l/	heta_2} = \frac{	heta_2}{	heta_1}$.Substituting the values: $\frac{r_1}{r_2} = \frac{2\pi/3}{5\pi/12} = \frac{2}{3} 	imes \frac{12}{5} = \frac{8}{5}$.Thus,the ratio is $8:5$.

If the arcs of the same length in two circles $S_1$ and $S_2$ subtend angles $75^{\circ}$ and $120^{\circ}$ respectively at the center,then the ratio of their radii $\frac{r_1}{r_2}$ is equal to:

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