Question

The two vertices of a triangle are $(4,2,1)$ and $(5,1,4)$. If the centroid is $(5,2,3)$, then the third vertex is

• A. $(3,4,5)$
• B. $(6,2,3)$
• C. $(6,3,2)$
• D. $(6,3,4)$

Answer 1

Answer: (D)

$(6,3,4)$

Answer 2

Centroid is $\left(\frac{\sum x_{i}}{3}, \frac{\sum y_{i}}{3}, \frac{\sum z_{i}}{3}\right)$ Let the third vertex be $\left(\alpha, \beta, \gamma\right)$. $\therefore 5=\frac{4+5+\alpha}{3}$ $\Rightarrow\alpha=6$, $2=\frac{2+1+\beta}{3}$ $\Rightarrow \beta=3, 3=\frac{1+4+\gamma}{3}$ $\Rightarrow \gamma=4$ $\therefore$ The third vertex is $\left(6 , 3 ,4\right)$.