Question

If $D,E$ and $F$ are respectively the mid-points of $AB,AC$ and $BC$ in $△ABC$, then $BE+AF$ is equal to

• A. $DC$
• B. $\frac{1}{2}BF$
• C. $2BF$
• D. $\frac{3}{2}BF$

Answer 1

Answer: (A)

$DC$

Let us represent $A,B$ and $C$ with vectors $\overline{a},\overline{b}$ and $\overline{c}$ respectively.

Here vectors representing the mid-points $D,E$ and $F$ are $\frac{\overline{a+b}}{2},\frac{\overline{a+c}}{2}$ and $\frac{\overline{b+c}}{2}$ respectively.

$\overline{BE}=\frac{\overline{a-2b+c}}{2}\overline{AF}=\frac{\overline{b-2a+c}}{2}\overline{CD}=\frac{\overline{a+b-2c}}{2}$

$\overline{BE}+\overline{AF}=\frac{\overline{2c-a-b}}{2}=-\overline{CD}=\overline{DC}$

Hence the option A is correct.