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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 other circle. Is this statement false ? Why ?
Given: 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 other circle.
To do: To check whether the statement is true or false? and to explain.
Solution:
Consider two circles $C_1$ and $C_2$ of radii $r$ and $2r$ respectively.
Let the length of two arcs be $l_1$ and $l_2$.
$l_1=\frac{2\pi r\theta_1}{360^o}$
$l_2=\frac{2\pi .2r\theta_2}{360^o}$
According $l_1=l_2$
$\Rightarrow \frac{2\pi r\theta_1}{360^o}=\frac{2\pi .2r\theta_2}{360^o}$
$\Rightarrow \theta_1=2\theta_2$
Angle of sector of the 1st circle is twice the angle of the sector of the other circle.
Hence, the given statement is true.
Updated on: 10-Oct-2022
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