Question

If the vectors $\overline{AB}=-3\overline{i}+4\overline{k}$ and $\overline{AC}=5\overline{i}-2\overline{j}+4\overline{k}$ are the sides of a $ABC$, then the length of the median through $A$ is

• A. $\sqrt{14}$
• B. $\sqrt{18}$
• C. $\sqrt{25}$
• D. $\sqrt{29}$

Answer 1

The correct option is B $\sqrt{18}$

$\overrightarrow{AB}=-3\hat{i}+4\hat{k}$

$\overrightarrow{AC}=5\hat{i}+2\hat{j}+4\hat{k}$

$\therefore \overrightarrow{BC}=\overrightarrow{AC}-\overrightarrow{AB}$

$=8\hat{i}-2\hat{j}$

Now as we know that in a triangle $2(A{D}^2+B{D}^2)=A{B}^2+A{C}^2$ where $D$ is the median,

By apollonius theorem.

$\therefore 2[{\overrightarrow{AD}}^2+[\frac{\left(\overrightarrow{BC}\right)}{2}{]}^2]={\overrightarrow{AB}}^2+{\overrightarrow{AC}}^2$

$\Rightarrow 2[l^2+17]=\left(25\right)+(25+4+16)$

$\Rightarrow l^2+17=25+2+8$

$\Rightarrow l^2=35-17$

$\Rightarrow l=\sqrt{18}$