Question

$A,B,C$ and $D$ have position vectors $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ and $\overrightarrow{d}$, respectively, such that $\overrightarrow{a}-\overrightarrow{b}=2(\overrightarrow{d}-\overrightarrow{c})$, then

• A. $AB$ and $CD$ bisect each other
• B. $AB$ and $CD$ trisect each other
• C. $BD$ and $AC$ bisect each other
• D. $BD$ and $AC$ trisect each other

Answer 1

The correct option is D $BD$ and $AC$ trisect each other

Given that

$\overrightarrow{a}-\overrightarrow{b}=2(\overrightarrow{d}-\overrightarrow{c})$

$\therefore \frac{\overrightarrow{a}+2\overrightarrow{c}}{2+1}=\frac{\overrightarrow{b}+2\overrightarrow{d}}{2+1}$

Hence, $AC$ and $BD$ trisect each other as $L.H.S$ is the position vector of a point trisecting $A$ and $C$, and $R.H.S$. that of $B$ and $D$.