Question

In the tetrahedron $ABCD,A=(1,2,-3)$ and $G(-3,4,5)$ is the centroid of the tetrahedron. If $P$ is the centroid of the $\Delta BCD$, then $AP=$

• A. $\frac{8\sqrt{21}}{3}$
• B. $\frac{4\sqrt{21}}{3}$
• C. $4\sqrt{21}$
• D. $\frac{\sqrt{21}}{3}$

Answer 1

The correct option is A $\frac{8\sqrt{21}}{3}$

Given, $A=(1,2,-3),G(-3,4,5)$

Therefore, $AG=\sqrt{{(-3-1)}^2+{(4-2)}^2+{(5-(-3\left)\right)}^2}$

and $AG=\sqrt{84}=2\sqrt{21}$

$P$ is the centroid of $△BCD$

So, $G$ divides $AP$ in $3:1$.

Let $AG=3x$, then $GP=x$

$3x=2\sqrt{21}x=\frac{2\sqrt{21}}{3}$

Now $AP=AG+GP$

$\Rightarrow AP=3x+x$

$\Rightarrow AP=4x$

$\Rightarrow AP=4\left(\frac{2\sqrt{21}}{3}\right)=\frac{8\sqrt{21}}{3}$

So, option A is correct.