Find the coordinates of the centroid of a triangle having vertices `P(x_1,y_1,z_1),Q(x_2,y_2,z_2) and R(x_3,y_3,z_3)`

Find the coordinates of the centroid of the triangle whose vertices are (x_(1),y_(1),z_(1))

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)) and (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))

Find the co oridinate of the centroid of the tetrahedron whose vertices are (x_(1),y_(1),z_(1)),(x_(2),y_(2),z_(2)),(x_(3),y_(3),z_(3)) and (x_(4),y_(4),z_(4))

STATEMENT-1 : The centroid of a tetrahedron with vertices (0, 0,0), (4, 0, 0), (0, -8, 0), (0, 0, 12)is (1, -2, 3). and STATEMENT-2 : The centroid of a triangle with vertices (x_(1), y_(1), z_(1)), (x_(2), y_(2), z_(2)) and (x_(3), y_(3), z_(3)) is ((x_(1)+x_(2)+x_(3))/3, (y_(1)+y_(2)+y_(3))/3, (z_(1)+z_(2)+z_(3))/3)

A tetrahedron is a three dimensional figure bounded by four non coplanar triangular plane.So a tetrahedron has four no coplnar points as its vertices. Suppose a tetrahedron has points A,B,C,D as its vertices which have coordinates (x_1,y_1,z_1)(x_2,y_2,z_2),(x_3,y_3,z_3) and (x_4,y_4,z_4) respectively in a rectangular three dimensional space. Then the coordinates of its centroid are ((x_1+x_2+x_3+x_3+x_4)/4, (y_1+y_2+y_3+y_3+y_4)/4, (z_1+z_2+z_3+z_3+z_4)/4) . the circumcentre of the tetrahedron is the center of a sphere passing through its vertices. So, this is a point equidistant from each of the vertices of the tetrahedron. Let a tetrahedron have three of its vertices represented by the points (0,0,0) ,(6,-5,-1) and (-4,1,3) and its centroid lies at the point (1,2,5). The coordinate of the fourth vertex of the tetrahedron is

Theorem : The area of a triangle the coordinates of whose vertices are (x_1;y_1);(x_2;y_2)and (x_3;y_3) is 1/2|(x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|

Let x_1y_1z_1,x_2y_2z_2 and x_3y_3z_3 be three 3-digit even numbers and Delta=|{:(x_1,y_1,z_1),(x_2,y_2,z_2),(x_3,y_3,z_3):}| . then Delta is

Planes r drawn parallel to the coordinate planes through the point P(x_(1),y_(1),z_(1) and Q(x_(2),y_(2),z_(2)). Find the length of the edges of the parallelepiped so formed.

The solution of the system of equations a_(1)x+b_(1)y+c_(1)z=d_(1),a_(2)x+b_(2)y+c_(2)z=d_(2) and a_(3)x+b_(3)y+c_(3)z=d_(3) using Crammer's rule is x=(Delta_(1))/(Delta),y=(Delta_(2))/(Delta) and z=(Delta_(3))/(Delta) where Delta!=0 The solution of 2x+y+z=1,x-2y-3z=1,3x+2y+4z=5 is