\(BC=\sqrt{AB^2+AC^2}=20\left(cm\right)\left(pytago\right)\)
Áp dụng HTL tam giác
\(\left\{{}\begin{matrix}AB^2=BH\cdot BC\\AC^2=CH\cdot BC\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}BH=\dfrac{AB^2}{BC}=\dfrac{144}{20}=7,2\left(cm\right)\\CH=\dfrac{AC^2}{BC}=\dfrac{256}{20}=12,8\left(cm\right)\end{matrix}\right.\)
Vì AD là phân giác nên \(\dfrac{AB}{AC}=\dfrac{BD}{DC}=\dfrac{12}{16}=\dfrac{3}{4}\Rightarrow BD=\dfrac{3}{4}DC\)
Mà \(BD+DC=BC=20\Leftrightarrow\dfrac{7}{4}DC=20\Leftrightarrow DC=\dfrac{80}{7}\left(cm\right)\)
\(\Leftrightarrow HD=CH-CD=12.8-\dfrac{80}{7}=\dfrac{48}{35}\left(cm\right)\)
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