The area of incircle of an equilateral triangle is $154 \mathrm{~cm}^{2}$. The perimeter of the triangle is
(a) 71.5 cm
(b) 71.7 cm
(c) 72.3 cm
(d) 72.7 cm
Area of incircle of equilateral triangle is $154 \mathrm{~cm}^{2}$
We have to find the perimeter of the triangle. So we will use area to get,
Area of incircle $=154$
$\pi r^{2}=154$
$r=\sqrt{\frac{154}{\pi}} \mathrm{cm}$
As triangle is equilateral so,
$\angle \mathrm{OCM}=30^{\circ}$
So,
$\tan 30^{\circ}=\frac{r}{\mathrm{MC}}$
$\mathrm{MC}=\sqrt{\frac{154(3)}{\pi}} \mathrm{cm}$
So,
$\mathrm{AC}=2(\mathrm{MC})$
$=2\left(\sqrt{\frac{154(3)}{\pi}}\right) \mathrm{cm}$
Therefore perimeter of the triangle is,
$=3(\mathrm{AC})$
$=6\left(\sqrt{\frac{462}{3.14}}\right)$
$=72.7 \mathrm{~cm}$
Therefore the answer is (d).
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