HELP

What is the radius of the circle inscribed in triangle ABC if AB=22, AC=12, BC=14?

Express your answer in simplest radical form.

\(\text{Let $c = AB = 22 $} \\ \text{Let $b = AC = 12 $} \\ \text{Let $a = BC = 14 $} \\ \text{Let $r =$ radius of the circle inscribed. }\)

Formula:

\(\begin{array}{|rcll|} \hline \displaystyle r = \sqrt{ \dfrac{(s-a)(s-b)(s-c)}{s} } \qquad \text{ with } \qquad s=\dfrac{a+b+c}{2} \\ \hline \end{array}\)

\(\mathbf{s = \ ?}\)

\(\begin{array}{|rcll|} \hline s &=& \dfrac{14+12+22}{2} \\\\ &=& \dfrac{48}{2} \\\\ &=& 24 \\ \hline \end{array} \)

\(\begin{array}{|rcll|} \hline s-a &=& 24- 14 \\ &=& 10 \\\\ s-b &=& 24 - 12 \\ &=& 12 \\\\ s-c &=& 24 - 22 \\ &=& 2 \\ \hline \end{array}\)

\(\mathbf{r = \ ?} \)

\(\begin{array}{|rcll|} \hline r &=& \sqrt{ \dfrac{(s-a)(s-b)(s-c)}{s} } \\\\ &=& \sqrt{ \dfrac{10\cdot 12 \cdot 2}{24} } \\\\ &=& \sqrt{ \dfrac{10\cdot 24}{24} } \\\\ &=& \sqrt{ 10 } \\ \hline \end{array}\)

The radius of the cricle inscribed in triangle ABC is \(\sqrt{10}\)