Question

Find AD, BF and CF in the figure.

Answer 1

Tangents draw from an external point to a circle are equal

$\begin{array}{c}\Rightarrow AD=AF,BD=BE,CE=CF.\\ LetAD=AF=a\\ BD=BE=b\\ CE=CF=c\\ AB=AD+DB=a+b=\mathrm{8.......}\left(1\right)\\ BC=BE+EC=b+c=\mathrm{10.......}\left(2\right)\\ AC=AF+FC=a+b=\mathrm{12.......}\left(3\right)\\ Adding\left(1\right),\left(2\right),and\left(3\right),weget\\ 2\left(a+b+c\right)=30\\ \Rightarrow \left(a+b+c\right)=\mathrm{15......}\left(4\right)\\ Subtracting\left(1\right)from\left(4\right),wegetc=7\\ Subtracting\left(2\right)from\left(4\right),wegeta=5\\ Subtracting\left(3\right)from\left(4\right),wegetb=3\\ Therefore,AD=a=5cm,BE=b=3cm,CF=c=7cm\end{array}$