Đáp án:
D
Giải thích các bước giải:
\(\overrightarrow {AM} = \overrightarrow {AB} + \overrightarrow {BM} = \overrightarrow {AB} + \dfrac{2}{3}\overrightarrow {BC} \) \( = \overrightarrow {AB} + \dfrac{2}{3}\left( {\overrightarrow {AC} - \overrightarrow {AB} } \right) = \dfrac{1}{3}\overrightarrow {AB} + \dfrac{2}{3}\overrightarrow {AC} \)
\( \Rightarrow A{M^2} = {\left( {\overrightarrow {AM} } \right)^2} = {\left( {\dfrac{1}{3}\overrightarrow {AB} + \dfrac{2}{3}\overrightarrow {AC} } \right)^2}\) \( = \dfrac{1}{9}A{B^2} + \dfrac{4}{9}A{C^2} + \dfrac{4}{9}\overrightarrow {AB} .\overrightarrow {AC} \)
\( = \dfrac{1}{9}{.6^2} + \dfrac{4}{9}{.4^2} + \dfrac{4}{9}.\left| {\overrightarrow {AB} } \right|.\left| {\overrightarrow {AC} } \right|.\cos \left( {\overrightarrow {AB} ,\overrightarrow {AC} } \right)\) \( = \dfrac{{148}}{9}\)
Do đó \(AM = \sqrt {\dfrac{{148}}{9}} = \dfrac{{2\sqrt {37} }}{3}\).
