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Show that the coordinates off the centro...

Show that the coordinates off the centroid of the triangle with vertices `A(x_1, y_1, z_1),\ B(x_2, y_2, z_2)a n d\ (x_3, y_3, z_3)` are `((x_1+x_2+x_3)/3,(y_1+y_2+y_3)/3,(z_1+z_2+z_3)/3)`

Text Solution

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Suppose D is the mid-point of BC. Then coordinates of D are
`((x_(2)+x_(3))/2, (y_(2)+y_(3))/2, (z_(2) +z_(3))/2)`

Let G the centroid of `triangle ABC.` Then G divides AD in the ratio 2 : 1.
Hence, coordinates of G are
`therefore ((1cdot x_(1) +2((x_(2)+x_(3))/2))/(1+2), (1 cdot y +2((y_(2)+y_(3))/2))/(1+2),(1 cdotz_(1)+2((z_(2)+z_(3))/2))/(1+2))`
`=((x_(1)+x_(2)+x_(3))/3, (y_(1)+y_(2)+y_(3))/3, (z_(1)+z_(2)+z_(3))/3)`
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