In the given figure, PT is a common tangent to three circles at points A, B and C respectively. The radius of the small, medium and large circles is 4 cm, 6 cm and 9 cm. O₁, O₂ and O₃ are the centre of the three circles what is the value (in cm) of PC?

Solution (By Examveda Team)

ΔPO₁A is similar to ΔPO₂B
AO₁ : BO₂ = 2 : 3
O₁O₂ = 10 = 3 - 2 = 1 unit
            1 unit → 10
PO₂ = 3 units → 30
ΔPO₃C is similar to ΔPO₂B.
BO₂ : CO₃ = 2 : 3
2 units → 30
1 unit → 15
PO₃ = 45
CO₃ = 9
$$\eqalign{ & {\text{PC}} = \sqrt {{{45}^2} - {9^2}} \cr & = \sqrt {2025 - 81} \cr & = \sqrt {1944} \cr & = 18\sqrt 6 \cr} $$