Question

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 ?

• A. True
• B. False

Answer 1

The correct option is A True

Consider two circles $C_1$ and $C_2$ of radii r and 2r respectively.
Let the length of two arcs be $l_1$ and $l_2.$

$l_1=\widehat{AB}$ of $C_1=\frac{2\pi r{\theta }_1}{{360}^{\circ }}$

$l_2=\widehat{CD}$ of $C_2=\frac{2\pi r{\theta }_2}{{360}^{\circ }}=\frac{2\pi .2r{\theta }_2}{{360}^{\circ }}$
According
$l_1=l_2$

$\Rightarrow \frac{2\pi r{\theta }_1}{{360}^{\circ }}=\frac{2\pi .2r{\theta }_2}{{360}^{\circ }}$

$\Rightarrow {\theta }_1=2{\theta }_2$
i.e., Angle of sector of the 1st circle is twice the angle of the sector of the other circle.
Hence, the given statement is true.