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

Answer 1

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

Given, $A=(1,2,-3),G(-3,4,5)$
$AG=\sqrt{(-3-1{)}^2+(4-2{)}^2+(5+3{)}^2}$
$AG=\sqrt{16+4+64}$
$AG=\sqrt{84}$
$P$ is the centroid of the $\Delta BCD$
So, $G$ divides $AP$ in $3:1$.
Let $AG=3x$, then $GP=x$
$3x=\sqrt{84}$
$3x=2\sqrt{21}$
$x=\frac{2\sqrt{21}}{3}$
Now $AP=AG+GP$
$AP=4x$
$AP=4\left(\frac{2\sqrt{21}}{3}\right)=\frac{8\sqrt{21}}{3}$