Đáp án:
\[\left\{ \begin{array}{l}
AC = \sqrt {14} \\
AB = \sqrt {30}
\end{array} \right.\]
Giải thích các bước giải:
Áp dụng công thức đường trung tuyến ta có:
\(\begin{array}{l}
\left\{ \begin{array}{l}
{m_b} = 4\\
{m_c} = 2\\
a = 3
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
{m_b}^2 = 16\\
{m_c}^2 = 4\\
a = 3
\end{array} \right.\\
\Leftrightarrow \left\{ \begin{array}{l}
\frac{{{a^2} + {c^2}}}{2} - \frac{{{b^2}}}{4} = 16\\
\frac{{{a^2} + {b^2}}}{2} - \frac{{{c^2}}}{4} = 4\\
a = 3
\end{array} \right.\\
\Leftrightarrow \left\{ \begin{array}{l}
\frac{{{3^2} + {c^2}}}{2} - \frac{{{b^2}}}{4} = 16\\
\frac{{{3^2} + {b^2}}}{2} - \frac{{{c^2}}}{4} = 4
\end{array} \right.\\
\Leftrightarrow \left\{ \begin{array}{l}
\frac{{{c^2}}}{2} - \frac{{{b^2}}}{4} = \frac{{23}}{2}\\
\frac{{{b^2}}}{2} - \frac{{{c^2}}}{4} = - \frac{1}{2}
\end{array} \right.\\
\Leftrightarrow \left\{ \begin{array}{l}
{b^2} = 14\\
{c^2} = 30
\end{array} \right.\\
\Leftrightarrow \left\{ \begin{array}{l}
b = AC = \sqrt {14} \\
c = AB = \sqrt {30}
\end{array} \right.
\end{array}\)
Vậy \(\left\{ \begin{array}{l}
AC = \sqrt {14} \\
AB = \sqrt {30}
\end{array} \right.\)
