Đáp án:
Giải thích các bước giải:
Câu 12:
\(\begin{array}{l}
\overrightarrow {MH} = - \overrightarrow {HM} = - \frac{1}{2}(\overrightarrow {HB} + \overrightarrow {HC} )\\
\overrightarrow {MA} = - \overrightarrow {AM} = \frac{{ - 1}}{2}\left( {\overrightarrow {AB} + \overrightarrow {AC} } \right)\\
\to \overrightarrow {MH} .\overrightarrow {MA} = - \frac{1}{2}(\overrightarrow {HB} + \overrightarrow {HC} ).\frac{{ - 1}}{2}\left( {\overrightarrow {AB} + \overrightarrow {AC} } \right)\\
= \frac{1}{4}\left( {\overrightarrow {HB} .\overrightarrow {AB} + \overrightarrow {HB} .\overrightarrow {AC} + \overrightarrow {HC} .\overrightarrow {AB} + \overrightarrow {HC} .\overrightarrow {AC} } \right)\\
= \frac{1}{4}(\overrightarrow {CH} .\overrightarrow {AC} + \overrightarrow {BH} .\overrightarrow {AB} ) ( do H là trực tâm)
\end{array}\)
\(\begin{array}{l}
= \frac{1}{4}\left[ {\overrightarrow {CH} .\left( {\overrightarrow {AB} + \overrightarrow {BC} } \right) + \overrightarrow {BH} .\left( {\overrightarrow {AB} + \overrightarrow {BC} } \right)} \right]\\
= \frac{1}{4}\left[ {\overrightarrow {CH} .\overrightarrow {AB} + \overrightarrow {CH} .\overrightarrow {BC} + \overrightarrow {BH} .\overrightarrow {AB} + \overrightarrow {BH} .\overrightarrow {BC} } \right]\\
= \frac{1}{4}\left( {\overrightarrow {CH} .\overrightarrow {BC} + \overrightarrow {BH} .\overrightarrow {BC} } \right)
\end{array}\)
(Do H là trực tâm)
\( = \frac{1}{4}\overrightarrow {BC} .(\overrightarrow {BH} + \overrightarrow {CH} ) = \frac{1}{4}\overrightarrow {BC} .\overrightarrow {BC} = \frac{1}{4}B{C^2}\)
