[tex] \frac{A_{ADB}}{A_{ADC}}= \frac{ \frac{1}{2} \cdot AD \cdot BD }{ \frac{1}{2} \cdot AD \cdot CD } = \frac{BD}{CD}= \frac{9}{16}. \\ \\ \frac{BD}{CD}= \frac{9}{16} \Rightarrow 16BD=9CD. \\ \\ BD+CD=BC=25 \\ \\ (16BD)+16CD=16 \cdot 25 \Leftrightarrow 6CD+16CD=16 \cdot25 \\ \\ Deci~25CD=16 \cdot 25 \Rightarrow CD=16~(cm). \\ \\ BD=BC-CD=25-16=9~(cm). [/tex]
[tex]Din~teorema~inaltimii~avem:~ \\ \\ AD^2= BD \cdot CD \Leftrightarrow AD^2=9 \cdot 16 \Leftrightarrow AD^2=3^2 \cdot 4^2 \Rightarrow \\ \\ \Rightarrow \boxed{AD=12~cm}~.[/tex]
