Question

Question 8
If the length of an arc of a circle of radius r is equal to that of an arc of a circle of radius 2r , then the angle of the corresponding sector of the first circle is double the angle of the corresponding sector of the other circle is this statement false? Why?

Answer 1

Let two circles $C_1$ and $C_2$ of radius r and 2r with centres O and $O^{{}^{\prime }}$ respectively.
It is given that, the arc length $\widehat{AB}$ of $C_1$ is equal to arc of length $\widehat{CD}$ of $C_2$ i.e. $\widehat{AB}=\widehat{CD}=I$ (say).
Now let ${\theta }_1$ be the angle subtended by arc $\widehat{AB}$ and ${\theta }_2$ be the angle subtended by arc $\widehat{CD}$ at the centre.

$\therefore \widehat{AB}=I=\frac{{\theta }_1}{360}\times 2\pi r$

$\widehat{CD}=I=\frac{{\theta }_2}{360}\times 2\pi \left(2r\right)=\frac{{\theta }_2}{360}\times 4\pi r$

From Eqs. (i) and (ii)

$\frac{{\theta }_1}{360}\times 2\pi r=\frac{{\theta }_2}{360}\times 4\pi r$

$\Rightarrow {\theta }_1=2{\theta }_2$

i.e angle formed by sector of $C_1$ is double the angle formed by sector $C_2$ at centre.

Therefore, the given statement is true.