Question

In the given figure, three circles with centres A, B, C, respectively, touch each other externally. If AB = 5 cm, BC = 7 cm and CA = 6 cm, the radius of the circle with centre A is


(a) 1.5 cm
(b) 2 cm
(c) 2.5 cm
(d) 3 cm

Answer 1

(b) 2 cm

$$\begin{gathered} \mathrm{Given},AB=5\mathrm{cm},BC=7\mathrm{cm}andCA=6\mathrm{cm}. \\ \mathrm{Let},AR=AP=x\mathrm{}\mathrm{cm}. \\ BQ=BP=y\mathrm{}\mathrm{cm} \\ CR=CQ=z\mathrm{}\mathrm{cm} \\ \left(\mathrm{Since}\mathrm{}\mathrm{the}\mathrm{}\mathrm{length}\mathrm{}\mathrm{of}\mathrm{}\mathrm{tangents}\mathrm{}\mathrm{drawn}\mathrm{}\mathrm{from}\mathrm{an}\mathrm{external}\mathrm{}\mathrm{point}\mathrm{}\mathrm{are}\mathrm{}\mathrm{equal}\right) \\ \mathrm{Then},AB=5\mathrm{}\mathrm{cm} \\ \Rightarrow AP+PB=5\mathrm{cm} \\ \Rightarrow x+y=5\dots ..\left(i\right) \\ \mathrm{Similarly},y+z=7\dots \dots \dots \left(ii\right) \\ \mathrm{and}z+x=6\dots .\left(iii\right) \\ \mathrm{Adding}\mathrm{}\left(\mathrm{i}\right),\left(\mathrm{ii}\right)\mathrm{}\mathrm{and}\left(\mathrm{iii}\right),\text{we get:} \\ \left(x+y\right)+\left(y+z\right)+\left(z+x\right)=18 \\ \Rightarrow 2\left(x+y+z\right)=18 \\ \Rightarrow \left(x+y+z\right)=9\dots \dots .\left(iv\right) \\ Now,\left(iv\right)-\left(ii\right): \\ \left(x+y+z\right)-\left(y+z\right)=\left(9-7\right) \\ \Rightarrow x=2 \\ \therefore \text{T}\mathrm{he}\mathrm{}\mathrm{radius}\mathrm{}\mathrm{of}\mathrm{}\mathrm{the}\mathrm{}\mathrm{circle}\mathrm{}\mathrm{with}\mathrm{}\mathrm{centre}\mathrm{}\mathrm{A}\mathrm{}\mathrm{is}\mathrm{}2\mathrm{cm}. \end{gathered}$$