# Prove the following

Question:

In $\Delta \mathrm{ABC}$, the lengths of sides $\mathrm{AC}$ and $\mathrm{AB}$ are $12 \mathrm{~cm}$ and $5 \mathrm{~cm}$, respectively. If the area of $\triangle \mathrm{ABC}$ is $30 \mathrm{~cm}^{2}$ and $\mathrm{R}$ and $\mathrm{r}$ are respectively the radii of circumcircle and incircle of $\triangle \mathrm{ABC}$ then the value of $2 R+r$ (in $c m$ ) is equal to____________.

Solution:

$\Delta=\frac{1}{2} \cdot 5 \cdot 12 \cdot \sin \mathrm{A}=30$

$\sin \mathrm{A}=1$

$\mathrm{A}=90^{\circ} \Rightarrow \mathrm{BC}=13$

$\mathrm{BC}=2 \mathrm{R}=13$

$\mathrm{r}=\frac{\Delta}{\mathrm{S}}=\frac{30}{15}=2$

$2 \mathrm{R}+\mathrm{r}=15$