Question

The area of a circle inscribed in an equilateral triangle is 154 cm^2.find the perimeter of the triangle

Answer 1

Given area of inscribed circle = 154 sq cm

Let the radius of the incircle be r.

⇒ Area of this circle = πr2 = 154

(22/7) × r2 = 154

⇒ r2 = 154 × (7/22) = 49

∴ r = 7 cm

Recall that incentre of a circle is the point of intersection of the angular bisectors.

Given ABC is an equilateral triangle and AD = h be the altitude.

Hence these bisectors are also the altitudes and medians whose point of intersection divides the medians in the ratio 2 : 1

∠ADB = 90° and OD = (1/3) AD

That is r = (h/3)

Þ h = 3r = 3 × 7 = 21 cm

Let each side of the triangle be a, then

Altitude of an equilateral triangle is (√3/2) times its side

That is h = (√3a/2)

∴ a = 14√3 cm

We know that perimeter of an equilateral triangle = 3a

= 3 × 14 √3 = 42√3

= 42 × 1.73 = 72.66 cm