Question

If the arcs of the same lengths in two circles subtend angles ${65}^{\circ }$ and ${110}^{\circ }$ at the centre, find the ratio of their radii.

Answer 1

Step 1: Given data
Let the radius of the two circles be $r_1$ and $r_2$.



Step 2: Solve for length of both circles
Length of arc of $1^{st}$ circle,
$l=r_1\theta =r_1\times {65}^{\circ }=r_1\times {65}^{\circ }\times \frac{\pi }{180}=r_1\times \frac{13\pi }{36}$
Length of arc of $2^{nd}$ circle,
$l=r_2\theta =r_2\times {110}^{\circ }=r_2\times {110}^{\circ }\times \frac{\pi }{180}=r_2\times \frac{11\pi }{18}$

Step 3: Ratio of radius
Given, length of $1^{st}$ arc = length of $2^{nd}$ arc
$\Rightarrow r_1\times \frac{13\pi }{36}=r_2\times \frac{11\pi }{18}$
$\Rightarrow \frac{r_1}{r_2}=\frac{11\pi }{18}\times \frac{36}{13\pi }$
$\Rightarrow \frac{r_1}{r_2}=\frac{22\pi }{13\pi }$
$\Rightarrow \frac{r_1}{r_2}=\frac{22}{13}$