Question

$C_1$ and $C_2$ are two congruent circles with centre $O_1$ and $0_2$ respectively. Each circle passes through the centre of other circle. If the circumferences of each circle is $2cm$, then find the perimeter of shaded region in (cm)

Answer 1

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Since circumference of circle = $2\pi r$

$\Rightarrow 2=2\pi r$

$\Rightarrow r=1/\pi$cm for both circles

From symmetry, perpendicular AB to

$O_1{O}_2$ also bisects $O_1{O}_2$

As $O_1{O}_2=r=\frac{1}{\pi }$cm

$(O_1$ is center for $O_1$ and $O_2$ lies on its circumference)

$\Rightarrow O_{1}L=\frac{1}{2}\times \frac{1}{\pi }=\frac{1}{2\pi }cm$

and $O_{1}A=r=\frac{1}{\pi }cm$

$\Rightarrow$ In $△O_{1}AL$, $cos\theta =\frac{O_{1}L}{O_{1}A}=\frac{1}{2\pi }\times \pi =\frac{1}{2}$

$\Rightarrow \theta ={60}^{\circ }=\frac{\pi }{3}$

As we know angle $=\frac{arc}{radius}$

for arc AB of circle $C_1$,

$arc=2\theta \times r=2\times \frac{\pi }{3}\times \frac{1}{\pi }=\frac{2}{3}cm$

Similarly arc AB of circle $C_2=\frac{2}{3}cm$

Perimeter of shaded region = $A{O}_{1}B+A{O}_{2}B=\frac{4}{3}cm$