Question

If centroid of the tetrahedron OABC, where coordinates of $A,B,C$ are $(a,2,3),(1,b,3)$ and $(2,1,c)$ respectively be $(1,2,3),$ then find the distance of a point $(a,b,c)$ from the origin, where O is the origin.

• A. $\sqrt{50}$
• B. $\sqrt{76}$
• C. 7
• D. $\sqrt{75}$

Answer 1

The correct option is D

$\sqrt{75}$

$A(x_1,y_1,z_1),B(x_2,y_2,z_2),C(x_3,y_3,z_3)$ and $D(x_4,y_4,z_4)$ are the vertices of a tetrahedron,then cooridanate of its centroid (G) is given as

$G(\frac{{\sum }_{1=1}^4{x}_1}{4},\frac{{\sum }_{1=1}^4{y}_1}{4},\frac{{\sum }_{1=1}^4{z}_1}{4})$

$1=\frac{0+a+1+2}{4},2=\frac{0+2+b+1}{4},3=\frac{0+3+2+c}{4}$

$a=1,b=5,c=7$

Point $(a,b,c)\equiv (1,5,7)$

Distance between the point (1,5,7) and origin is

$=\sqrt{1^2+5^2+7^2}$

$=\sqrt{1+25+49}$

$=\sqrt{75}$

Option D is correct