Question

Cho hình chữ nhật ABCD tâm O, I trung điểm BO. CMR 3AB\(\overrightarrow{ }\) + AD\(\overrightarrow{ }\)\(=\)4AI\(\overrightarrow{ }\)

Answer 1

Do I là trung điểm OB \(\Rightarrow\left\{{}\begin{matrix}\overrightarrow{IB}=\frac{1}{2}\overrightarrow{OB}\\\overrightarrow{ID}=\overrightarrow{IO}+\overrightarrow{OD}=\frac{1}{2}\overrightarrow{OD}+\overrightarrow{OD}=\frac{3}{2}\overrightarrow{OD}\end{matrix}\right.\)

\(\Rightarrow\overrightarrow{ID}=-3\overrightarrow{IB}\Rightarrow3\overrightarrow{IB}+\overrightarrow{ID}=\overrightarrow{0}\)

Do đó:

\(3\overrightarrow{AB}+\overrightarrow{AD}=3\left(\overrightarrow{AI}+\overrightarrow{IB}\right)+\overrightarrow{AI}+\overrightarrow{ID}\)

\(=4\overrightarrow{AI}+3\overrightarrow{IB}+\overrightarrow{ID}=4\overrightarrow{AI}\)