Mathematics - 3D Coordinate Geometry Question with Solution | TestHub

The coordinates of the centroid of a tetrahedron with vertices $(x_1,y_1,z_1)$, $(x_2,y_2,z_2)$, $(x_3,y_3,z_3)$, and $(x_4,y_4,z_4)$ are given by $\left(\frac{x_1+x_2+x_3+x_4}{4},\frac{y_1+y_2+y_3+y_4}{4},\frac{z_1+z_2+z_3+z_4}{4}\right)$.

Given the vertices $O(0,0,0)$, $A(a,2,3)$, $B(1,b,2)$, and $C(2,1,c)$, and the centroid $(1,2,-1)$, equating the coordinates gives:

$\frac{0+a+1+2}{4}=1\Rightarrow a+3=4\Rightarrow a=1$

$\frac{0+2+b+1}{4}=2\Rightarrow b+3=8\Rightarrow b=5$

$\frac{0+3+2+c}{4}=-1\Rightarrow c+5=-4\Rightarrow c=-9$

The point $P(a,b,c)$ is $(1,5,-9)$.

The distance of $P$ from the origin is:

$OP=\sqrt{1^2+5^2+(-9{)}^2}=\sqrt{1+25+81}=\sqrt{107}$ units.

Answer: $\sqrt{107}$ units