Question

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

Answer 1

Given :

The area of the circle inscribed in an equilateral triangle ABC is 154 cm².

To Find :

Perimeter of the triangle

Solution :

Let the radius of the circle be r cm.

Area of circle :

=> πr² = 154

=> 22/7 × r² = 154

=> r² = 49

=> r = 7 cm

Since ABC is an equilateral triangle.

therefore ∠OBM = ∠ABC/2 = 60/2 = 30 degree

and M will be the mid-point of BC.

In the triangle OBM

tan 30 = OM/BM

1/$\sqrt{3}$ = r/BM

BM = $\sqrt{3}$r

BM = 7$\sqrt{3}$

Therefore the length of the side of the triangle

=> BC = 2BM

=> BC = 14 $\sqrt{3}$

Thus perimeter of triangle = 3BC

= 3×14$\sqrt{3}$ = 42$\sqrt{3}$ cm

Answer 2

ɢɪᴠᴇɴ

• ᴀʀᴇᴀ ᴏғ ɪɴsᴄʀɪʙᴇᴅ ᴄɪʀᴄʟᴇ ɪɴ ᴇǫᴜɪʟᴀᴛᴇʀᴀʟ ᴛʀɪᴀɴɢʟᴇ = 154cm²

ᴛᴏ ғɪɴᴅ

• ᴘᴇʀɪᴍᴇᴛᴇʀ ᴏғ ᴛʜᴇ ᴛʀɪᴀɴɢʟᴇ.

sᴏʟᴜᴛɪᴏɴ

Let the radius of inscribed circle be be =r

Then,

Area Of circle = 154cm²

➲ π × r² = 154

22/7 × r² = 154

r² = 154 ×7/22

r² =7 ×7

r² = 49

r = √49

r = 7 cm

Now,

Let the height of triangle be h

r = h/3

h = 3r

h= 3×7

h = 21 cm

If a is the side of the triangle Then

h = √3/2 ×a

a = 2h/√3

a = 21×2/√3

a = 42/√3

a = 14√3 cm

Hence,The perimeter of triangle =3a

Perimeter = 3×14√3

Perimeter = 3× 14× 1.73

Perimeter= 72.7 cm

∴The Perimeter of triangle is 72.7cm