If the arcs of the same length in two circles subtend angles $65{}^\circ $ and $110{}^\circ $ at the centres of the circles. Find the ratio of the radii of the circles.
VerifiedHint: Use the fact that if the angle subtended at the centre of a circle of the radius by the arc of the circle of length l is x in degrees, then we have $l=\dfrac{x}{360}\times 2\pi r$. Hence find the lengths of arc in the two cases. Equate the length of the arcs and hence find the ratio of the radii of the circles.
Complete step-by-step answer:
Let ${{r}_{1}}$ be the radius of the first circle and ${{r}_{2}}$ the radius of the second circle. Let l be the length of the arcs.
Now, we know that if the angle subtended at the centre of a circle of the radius by the arc of the circle of length l is x in degrees, then we have $l=\dfrac{x}{360}\times 2\pi r$.
Hence, we have
$l=\dfrac{65}{360}\,\times 2\pi {{r}_{1}}\text{ - (i)}$
And $l=\dfrac{110}{360}\times 2\pi {{r}_{2}}$-(ii)
Hence from equation (i) and (ii), we get
$\dfrac{65}{360}\times 2\pi {{r}_{1}}=\dfrac{110}{360}\times 2\pi {{r}_{2}}$
Multiplying both sides of the equation by $\dfrac{360}{2\pi }$, we get
$65{{r}_{1}}=110{{r}_{2}}$
Dividing both sides by 65, we get
${{r}_{1}}=\dfrac{110}{65}{{r}_{2}}=\dfrac{22}{13}{{r}_{2}}$
Dividing both sides of the equation by ${{r}_{2}}$, we get
\[ \dfrac{{{r}_{1}}}{{{r}_{2}}}=\dfrac{22}{13} \]
\[ \Rightarrow {{r}_{1}}:{{r}_{2}}=22:13 \]
Hence the ratio of the radii is 22:13
Note: Alternative solution:
We know that in a circle, the angle subtended by the arc is directly proportional to the arc length and inversely proportional to the circle radius.
Hence, we have
$\theta \propto \dfrac{1}{r}$
Hence, we have
$\dfrac{{{r}_{1}}}{{{r}_{2}}}=\dfrac{{{\theta }_{2}}}{{{\theta }_{1}}}=\dfrac{110}{65}=22:13$
Hence the ratio of the radii of the circles is 22:13
