[tex]\displaystyle \\
\texttt{Se da: } \\
\Delta ABC ~~cu~~\ \textless \ A=90^o \\
BC = 40 cm \\ \\
\frac{AB}{AC} = \frac{3}{4} \\ \\
AD \perp BC ~~~D \in BC~~~(inaltime)\\
\texttt{Se cere: } \\
AB=? \\
AC=? \\
AD=? \\ \\
Rezolvare: \\ \\
\frac{AB}{AC} = \frac{3}{4} \\ \\
\Longrightarrow~~~AB =3k~~~si~~~ AC=4k \\
\texttt{Aplicam Pitagora: } \\
AB^2+AC^2 = BC^2 \\
(3k)^2 + (4k)^2=40^2 \\
9k^2 + 16K^2 = 1600 \\
k^2(9+16) = 1600 \\
25 k^2 = 1600 \\ \\
k^2 = \frac{1600}{25}[/tex]
[tex]\displaystyle \\
k = \sqrt{\frac{1600}{25}} =\frac{40}{5} = \boxed{8} \\ \\
AB = 3k = 3\times 8 = \boxed{24 ~cm} \\
AC = 4k = 4\times 8 = \boxed{32 ~cm} \\ \\
\texttt{Calculam inaltimea:} \\
BC \times AD =AB \times AC\\ \\
AD = \frac{AB \times AC}{BC}= \frac{24 \times 32}{40}= \frac{24 \times 8}{10}=\frac{192}{10}= \boxed{19,2 ~cm}[/tex]
