In the given figure, $\displaystyle \Delta \mathrm{ABC}$ is circumscribing a circle. Find the length of BC , if $\displaystyle \mathrm{AR}=4 \mathrm{~cm}, \mathrm{BR}=3 \mathrm{~cm}$ and $\displaystyle \mathrm{AC}=11 \mathrm{~cm}$.

\(\displaystyle \mathrm{BP}=\mathrm{BR}=3 \mathrm{~cm}\) and \(\displaystyle \mathrm{AQ}=\mathrm{AR}=4 \mathrm{~cm}\) \[\begin{aligned} & \mathrm{QC}=\mathrm{AC}-\mathrm{AQ}=11-4=7 \mathrm{~cm} \\ & \mathrm{PC}=\mathrm{QC}=7 \mathrm{~cm} \\ & \therefore \mathrm{BC}=\mathrm{BP}+\mathrm{PC}=3+7=10 \mathrm{~cm} \end{aligned} \]