Question

In the $\Delta ABC$, if $AB=\sqrt{2};AC=\sqrt{20},B=(3,2,0)$ and $C=(0,1,4)$, then the length of the median passing through $A$ is

• A. $\frac{3}{2}$
• B. $\frac{9}{2}$
• C. $\frac{3}{\sqrt{2}}$
• D. $\frac{\sqrt{3}}{2}$

Answer 1

The correct option is C $\frac{3}{\sqrt{2}}$

Using Apollonius theorem: In any triangle $ABC$, if $AD$ is a median,

then ${AB}^2+{AC}^2=2({BD}^2+{AD}^2)$
$D$ is the midpoint of $BC$
Coordinates of $D$ are $(\frac{3}{2},\frac{3}{2},2)$ and $(BD{)}^2=\frac{26}{4}$

$\Rightarrow {AB}^2+{AC}^2=2({BD}^2+{AD}^2)$

$\Rightarrow AD=\frac{3}{\sqrt{2}}$