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Find the third vertex of triangle whose centroid is origin and two vertices are (2, 4, 6) and (0, –2, –5).
Find the third vertex of triangle whose centroid is origin and two vertices are (2, 4, 6) and (0, –2, –5).
CBSE | Class 11 | Introduction to Three Dimensional Geometry | Maths | NCERT Exemplar
Find the third vertex of triangle whose centroid is origin and two vertices are (2, 4, 6) and (0, –2, –5).

Solution:

It is given the centroid is origin and two vertices are $(2,4,6)$ and $(0,-2,-5)$
Suppose the third vertex be $(x, y, z)$
The coordinates of the centroid for a triangle is given by the average of the coordinates of its vertices.
$\begin{array}{l}
\Rightarrow(0,0,0)=\left(\frac{2+0+x}{3}, \frac{4+(-2)+y}{3}, \frac{6+(-5)+z}{3}\right) \\
\Rightarrow \frac{2+x}{3}=0, \therefore x=-2 \\
\Rightarrow \frac{2+y}{3}=0, \therefore y=-2 \\
\Rightarrow \frac{1+x}{3}=0, \therefore x=-1
\end{array}$
So, the third vertex is $(-2,-2,-1)$.

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