Question

Cho hbh ABCD,M là trung điểm cạnh CD,N là trung điểm đoạn BM.

CMR:\(\overrightarrow{AN}=\frac{3}{4}\overrightarrow{AB}+\frac{1}{2}\overrightarrow{AD}\)

Answer 1

ta có \(2\overrightarrow{AN}=\overrightarrow{AB}+\overrightarrow{AM}=>2\overrightarrow{AN}=\overrightarrow{AB}+\overrightarrow{AD}+\overrightarrow{DM}\)

=>\(2\overrightarrow{AN}=\overrightarrow{AB}+\overrightarrow{AD}+\frac{1}{2}\overrightarrow{DC}\\ =>2\overrightarrow{AN}=\frac{3}{2}\overrightarrow{AB}+\overrightarrow{AD}\)

=>\(\overrightarrow{AN}=\frac{3}{4}\overrightarrow{AB}+\frac{1}{2}\overrightarrow{AD}\)

Answer 2

\(\overrightarrow{AN}=\frac{1}{2}\overrightarrow{AB}+\frac{1}{2}\overrightarrow{AM}=\frac{1}{2}\overrightarrow{AB}+\frac{1}{2}.\left(\frac{1}{2}\overrightarrow{AC}+\frac{1}{2}\overrightarrow{AD}\right)\)

\(=\frac{1}{2}\overrightarrow{AB}+\frac{1}{4}\overrightarrow{AC}+\frac{1}{4}\overrightarrow{AD}=\frac{1}{2}\overrightarrow{AB}+\frac{1}{4}\overrightarrow{AB}+\frac{1}{4}\overrightarrow{AD}+\frac{1}{4}\overrightarrow{AD}\)

\(=\frac{3}{4}\overrightarrow{AB}+\frac{1}{2}\overrightarrow{AD}\)