Question

If the centroid of a triangle whose vertices are $(a,1,3),(-2,b,-5)$ and $(4,7,c)$be the origin, then the values of $a,b$ and $c$ are:

• A. $-2,-8,-2$
• B. $2,8,-2$
• C. $-2,-8,2$
• D. $7,-1,0$

Answer 1

The correct option is C

$-2,-8,2$

Explanation for the correct option.

Let $\mathrm{P}\left({\mathrm{x}}_1,{\mathrm{y}}_1\right),\mathrm{Q}\left({\mathrm{x}}_2,{\mathrm{y}}_2\right)$ and $\mathrm{R}\left({\mathrm{x}}_3,{\mathrm{y}}_3\right)$ be coordinates of the triangle.
We know that the centroid of the triangle is given by $\left(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3}\right)$.
Substituting values we get:
$$\begin{gathered} (0,0,0)=\left(\frac{(a-2+4)}{3},\frac{(1+b+7)}{3},\frac{(3-5+c)}{3}\right) \\ \Rightarrow (0,0,0)=\left(\frac{(a+2)}{3},\frac{(b+8)}{3},\frac{(c-2)}{3}\right) \end{gathered}$$
Comparing coordinates we get:

$\frac{a+2}{3}=0,\frac{b+8}{3}=0\mathrm{and}\frac{c-2}{3}=0$

$\Rightarrow a=-2,b=-8,c=2$

Hence, option C is correct.