Question

$A=(1,-2,3)$ , $B=$ (2, 1, 3), $C=$ (4, 2, 1) and $G=(-1,3,5)$ is the centroid of the tetrahedron $ABCD$. Then the fourth coordinate is

• A. $(11,11,13)$
• B. $(-11,11,45)$
• C. $(-11,11,13)$
• D. $(11,13,11)$

Answer 1

Answer: (C)

$(-11,11,13)$

Given : $A=(1,-2,3),B=(2,1,3),C=(4,2,1)$ and $G=(-1,3,5)$

Let $D=(a,b,c)$
$G$ is centroid of the tetrahedron $ABCD$ then,

$G=\frac{A+B+C+D}{4}$

$⟹G=\frac{(1,-2,3)+(2,1,3)+(4,2,1)+(a,b,c)}{4}$

$⟹4G=(1+2+4+a,-2+1+2+b,3+3+1+c)$

$⟹(-4,12,20)=(7+a,1+b,7+c)$

$⟹7+a=-4,1+b=12,7+c=20$

$⟹a=-11,b=11,c=13$

Hence, $D=(-11,11,13)$ is the fourth coordinate.