Question

$I$: Axes are translated to $(1,0,-1)$ then the new coordinates of $(1,2,-3)$ are $(0,2,2)$.
$II$: If $(3,2,-1),(4,1,1)$ and $(6,2,5)$ are three vertices of a tetrahedron and $(4,2,2)$ is the centroid, then fourth vertex is $(3,3,3)$.

Which of the above statements is correct?

• A. Only I
• B. Only II
• C. Both I and II
• D. Neither I nor II

Answer 1

Answer: (B)

Only II

$I$. Axis is shifted to $(1,0,-1)$
Therefore, new coordinate will be $(X,Y,Z)\equiv (1-1,2-0,-3+1)\equiv (0,2,-2)$
$II$. Let the fourth vertex be $(a,b,c)$
By using the formula of centroid of tetrahedron, $(4,2,2)\equiv [\frac{3+4+6+a}{4},\frac{2+1+2+b}{4},\frac{-1+1+5+c}{4}]$
$(4,2,2)\equiv [\frac{a+13}{4},\frac{b+5}{4},\frac{c+5}{4}]$
On solving, we get
$a=3$, $b=3$ and $c=3$