Three circles touch each other externally. A triangle is formed when the centers of these circles are joined together. Find the radii of the circles, if the sides of the triangle formed are 6 cm, 8 cm and 9 cm.

Let there be three circles with center A, B and C and ABC be the triangle formed when centres of 3 circles are joined.

Let,

AB = 6 cm, AC = 8 cm and BC = 9 cm.

Let the radii of the circles having centres A, B and C be r₁, r₂ and r₃ respectively.

As,

⇒ AB = r₁ + r₂ = 6 …………(1)

⇒ AC = r₁ + r₃ = 8 …………(2)

⇒ BC = r₂ + r₃ = 9 ………….(3)

Adding all the above equations, we get

⇒ r₁ + r₂ + r₁ + r₃ + r₂ + r₃ = 6 + 8 + 9

⇒ 2(r₁ + r₂ + r₃) = 23

⇒ r₁ + r₂ + r₃ = $\dfrac{23}{2}$

⇒ r₁ + r₂ + r₃ = 11.5 cm ………..(4)

Substituting value of r₂ + r₃ = 9 in equation (4) we get :

⇒ r₁ + 9 = 11.5

⇒ r₁ = 11.5 - 9

⇒ r₁ = 2.5 cm.

Substituting value of r₁ + r₂ = 6 in equation (4) we get :

⇒ 6 + r₃ = 11.5

⇒ r₃ = 11.5 - 6

⇒ r₃ = 5.5 cm.

Substituting value of r₁ + r₃ = 8 in equation (4) we get :

⇒ 8 + r₂ = 11.5

⇒ r₂ = 11.5 - 8

⇒ r₂ = 3.5 cm.

Hence, the radii of the circles are 2.5 cm, 3.5 cm and 5.5 cm.