If the centroid of triangle whose vertices are (a, 1, 3), (–2, b, –5) and (4, 7, c) is origin, then the value of c – a – b is ___
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Solution

Coordinates of centroid of the triangle is given by
$G \equiv (\frac{x_{1} + x_{2} + x_{3}}{3}, \frac{y_{1} + y_{2} + y_{3}}{3}, \frac{z_{1} + z_{2} + z_{3}}{3})$
Substituting the coordinates of triangle in the above relation, we get
$G \equiv (\frac{a - 2 + 4}{3}, \frac{1 + b + 7}{3}, \frac{3 - 5 + c}{3})$
We are given centroid is the origin (0,0,0) .
$\Rightarrow$ $\frac{a - 2 + 4}{3} = 0, \frac{1 + b + 7}{3} = 0$ and $\frac{3 - 5 + c}{3} = 0$
$\frac{a - 2 + 4}{3} = 0 \Rightarrow a = - 2$
$\frac{1 + b + 7}{3} = 0 \Rightarrow b = - 8$
$\frac{3 - 5 + c}{3} = 0 \Rightarrow c = 2$
$\Rightarrow$ $c - a - b = 2 - (- 2) - (- 8)$
$= 2 + 2 + 8$
$= 12$

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