a: \(AH^2=HB\cdot HC\)
\(\Leftrightarrow HB\left(HB+9\right)=36\)
=>HB=3(cm)
=>HC=12cm
\(AC=\sqrt{12\cdot15}=5\sqrt{6}\left(cm\right)\)
\(AB=\sqrt{3\cdot15}=3\sqrt{5}\left(cm\right)\)
c: \(AB^2=BH\cdot BC\)
\(\Leftrightarrow BH\left(BH+9\right)-400=0\)
\(\Leftrightarrow\left(BH+25\right)\left(BH-16\right)=0\)
=>BH=16(cm)
\(AH=\sqrt{HB\cdot HC}=12\left(cm\right)\)
d: BC=BD+CD=17,5(cm)
Xét ΔABC có AD là phân giác
nên AB/DB=AC/DC
=>AB/3=AC/4=k
=>AB=3k; AC=4k
\(AC^2+AB^2=BC^2\)
\(\Leftrightarrow25k^2=17.5^2\)
\(\Leftrightarrow k^2=12.25\)
=>k=3,5
=>AB=10,5(cm); AC=14(cm)
\(AH=\dfrac{AB\cdot AC}{BC}=\dfrac{10.5\cdot14}{17.5}=8.4\left(cm\right)\)
\(BH=\dfrac{AB^2}{BC}=\dfrac{10.5^2}{17.5}=6.3\left(cm\right)\)
