Question

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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Answer 1

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 $\frac{3-5+c}{3}=0$
$\frac{a-2+4}{3}=0\Rightarrow a=-2$
$\frac{1+b+7}{3}=0\Rightarrow b=-8$
$\frac{3-5+c}{3}=0\Rightarrow c=2$
$\Rightarrow$​ $c-a-b=2-(-2)-(-8)$
$=2+2+8$
$=12$