Đáp án: BH = 4,5cm và CH = 8cm
Giải thích các bước giải:
$\begin{array}{l}
\frac{1}{{A{B^2}}} + \frac{1}{{A{C^2}}} = \frac{1}{{A{H^2}}} = \frac{1}{{36}}\\
\Rightarrow \frac{1}{{A{B^2}}} + \frac{1}{{{{\left( {\frac{4}{3}AB} \right)}^2}}} = \frac{1}{{36}}\\
\Rightarrow \frac{1}{{A{B^2}}} + \frac{9}{{16}}.\frac{1}{{A{B^2}}} = \frac{1}{{36}}\\
\Rightarrow \frac{1}{{A{B^2}}} = \frac{4}{{225}}\\
\Rightarrow AB = \frac{{15}}{2} = 7,5\left( {cm} \right)\\
\Rightarrow AC = \frac{4}{3}.7,5 = 10\left( {cm} \right)\\
\Rightarrow \left\{ \begin{array}{l}
BH = \sqrt {A{B^2} - A{H^2}} = 4,5\left( {cm} \right)\\
CH = \sqrt {A{C^2} - A{H^2}} = 8\left( {cm} \right)
\end{array} \right.
\end{array}$
