With the vertices of Δ PQR as centres, three circles are described, each touching the other two externally. If the sides of the triangle are 7 cm, 8 cm and 11 cm, find the radii of the three circles.

Let radius of circles with center P, Q and R be r₁, r₂ and r₃.

⇒ PQ = r₁ + r₂ = 7 …………(1)

⇒ PR = r₁ + r₃ = 8 …………(2)

⇒ QR = r₂ + r₃ = 11 ………….(3)

Adding all the above equations, we get

⇒ r₁ + r₂ + r₁ + r₃ + r₂ + r₃ = 7 + 8 + 11

⇒ 2(r₁ + r₂ + r₃) = 26

⇒ r₁ + r₂ + r₃ = $\dfrac{26}{2}$ ​ ⇒ r₁ + r₂ + r₃ = 13 cm ………..(4)

Substituting value of r₂ + r₃ = 11 in equation (4) we get :

⇒ r₁ + 11 = 13

⇒ r₁ = 13 - 11

⇒ r₁ = 2 cm.

Substituting value of r₁ + r₂ = 7 in equation (4) we get :

⇒ 7 + r₃ = 13

⇒ r₃ = 13 - 7

⇒ r₃ = 6 cm.

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

⇒ 8 + r₂ = 13

⇒ r₂ = 13 - 8

⇒ r₂ = 5 cm.

Hence, the radii of the circles with center P, Q and R are 2 cm, 5 cm and 6 cm.