Question

The area of a circle inscribed in an equilateral triangle is $154$ $c{m}^2$. Find the perimeter of the triangle.

• A. $46.3$ cm
• B. $66.3$ cm
• C. $72.7$ cm
• D. $78.8$ cm

Answer 1

The correct option is C $72.7$ cm

Area of a circle $=154c{m}^2$
$\Rightarrow \pi r^2=154$
$\Rightarrow r^2=\frac{154}{22}\times 7$
$\Rightarrow r=7$ cm
We know that in a equilateral triangle centroid and circumcentre coincides.
In above question point $B$ is the centroid and circumcentre of triangle $DCE$ and $B$ divides $AC$ in the ratio $1:2$.
$\Rightarrow AB=7$ (radius of circle)
$\Rightarrow BC=2\left(7\right)=14$
$\Rightarrow AC=14+7=21$ cm
Let side for triangle $x=DE=CE=CD$
$A$ is the mid point of $DE$
$\therefore DA=\frac{x}{2}$
Applying pythagoreus theorem on triangle $ADC=A{C}^2+D{C}^2=D{C}^2$
$\Rightarrow (21{)}^2+{\left(\frac{x}{2}\right)}^2=x^2$
$\Rightarrow 441+\frac{x^2}{4}=x^2$
$\Rightarrow \frac{3}{4}{x}^2=441$

$\Rightarrow x=\frac{42}{\sqrt{3}}$
Perimeter of equilateral triangle $=3\left(x\right)$
$=3\times \frac{42}{\sqrt{3}}=72.7$ cm