### A circle is inscribed in a $\triangle ABC$, touching $BC, CA,$ and $AB$ at points $P, Q,$ and $R$ respectively. If $AB = 14 \space cm, AQ = 11 \space cm$ and $CQ = 9 \space cm$ then find the length of $BC$. R Q P B C A

$12 \space cm$
Thus, \begin{aligned} & AR = AQ = 11 \space cm \\ & BP = BR \\ \text{ and } & CP = CQ = 9 \space cm \end{aligned}
2. We see that $BR = AB - AR = 14 \space cm - 11 \space cm = 3 \space cm.$
Thus, $BP = BR = 3 \space cm.$
3. Therefore, $BC = BP + CP = 3 \space cm + 9 \space cm = 12 \space cm.$