[tex]\displaystyle\\
\texttt{\bf Calculam aria triunghiului ABC.}\\\\
\text{Ducem inaltimea }AD\perp BC,~D\in BC\\
\text{In } \Delta ABD,~\text{dreptunghic in }D,~\text{avem:}\\
AB = 13~cm = \text{ipotenuza}\\
BD = \frac{BC}{2}= \frac{10}{2}=5~cm\\
\text{Cateta }AD = \sqrt{AB^2-BD^2}=\\
=\sqrt{13^2-5^2}=\sqrt{169-25}=\sqrt{144}=\boxed{12~cm}\\
A_{\Delta ABC} = \frac{BC\times AD}{2} = \frac{10\times 12}{2} = 10\times6 = \boxed{60~cm^2}[/tex]
[tex]\displaystyle\\
\texttt{\bf Calculam semiperimetrul triunghiului ABC.} \\\\
p = \frac{AB+AC+BC}{2}= \frac{13+13+10}{2}= \frac{36}{2}=\boxed{18~cm}\\\\\\
\texttt{\bf Calculam raza cercului inscris in triunghiul ABC.} \\\\
\texttt{\bf r} = \frac{A_{\Delta ABC}}{p} =\frac{60}{18} =\frac{10}{3} = \boxed{\bf 3,33~cm}[/tex]
