Question

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

• A. $\sqrt{45}$
• B. $\sqrt{18}$
• C. $\sqrt{72}$
• D. $\sqrt{33}$

Answer 1

Answer: (D)

$\sqrt{33}$

Take A as the origin(0,0)

so Vector $\overrightarrow{AB}=3\hat{i}+4\hat{k}and\overrightarrow{AC}=5\hat{i}-2\hat{j}+4\hat{k}$ will become position vector.

So coordinate of B is ($3,0,4$) and C is ($5,-2,4$)

The mid point of BC can be easily found by midpoint formula of two points as D($4,-1,4$).

so length of median through A is

$$\begin{array}{c}\overrightarrow{AD}=\sqrt{{\left(4-0\right)}^2+{\left(-1-0\right)}^2+{\left(4-0\right)}^2}\\ =\sqrt{33}\end{array}$$