Question

If $\left(\frac{9}{4}, \frac{5}{4}, \frac{15}{4}\right)$ is the centroid of a tetrahedron whose vertices are $(a, 2,1),(1, b, 4),(4,0, c)$ and $(1,1,7)$, then

• A. $a=b=c$
• B. $a=b=c+1$
• C. $b=c=a+1$
• D. $a=c=b+1$

Answer 1

Answer: (D)

Vertices of tetrahedron
$(a, 2,1),(1, b, 4),(4,0, c),(1,1,7)$
Centroid
$ \left(\frac{a+1+4+1}{4}, \frac{2+b+0+1}{4}, \frac{1+4+c+7}{4}\right) $
$ =\left(\frac{9}{4}, \frac{5}{4}, \frac{15}{4}\right) $
$ \Rightarrow \frac{a+6}{4}=\frac{9}{4}, \frac{b+3}{4}=\frac{5}{4}, \frac{c+12}{4}=\frac{15}{4} $
$ \Rightarrow a=3, b=2 c=3$
$ \Rightarrow a=c=b+1$