Question

The area of a circle inscribed in an equilateral triangle is 154 cm². Find the perimeter of the triangle.

Answer 1

Let the radius of the inscribed circle be r cm.
Given:
Area of the circle = 154 cm²

We know ; 

Area of the circle = πr² 

`=> 154 = 22/7"r"^2`

`=> (154xx7)/22 = "r"^2` 

⇒ r² = 49

⇒ r = 7

In a triangle, the centre of the inscribed circle is the point of intersection of the medians and altitudes of the triangle. The centroid divides the median of a triangle in the ratio 2:1.
Here,

AO;OD = 2:1

Now, 

Let the altitude be h cm

We have : 

∠ADB = 90°

OD `=1/3"AD"`

OD `= "h"/3`

⇒ h = 3r

⇒ h = 21

Let each side of the triangle be a cm.

In the right - angled ΔADB, we have ;

AB²  = AD²  + DB2

`"a"^2 = "h"^2 + (a/2)^2`

`4"a"^2  = 4"h"^2 + "a"^2`

`a^2 = 4"h"^2`

`a^2 = (4"h"^2)/3`

`a = (2"h")/sqrt(3)`

`a = 42/sqrt(3) ` 

∴  Perimeter of the triangle = 3a

`= 3xx42/sqrt(3)`

`= sqrt(3)xx42`

= 72.66 cm