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

A tetrahedron is three dimensional figure bounded by four non coplanar triangular plane. So 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_(4))/(4) , (y _(1) + y _(2) + y_(3) + y _(4))/(4), (z_(1) + z_(2) + z_(3)+ z_(4))/(4)). The circumcentre of the tetrahedron is the centre of a sphere passing through its vertices. So, the circumcentre is a point equidistant from each of the vertices of tetrahedron. Let tetrahedron has 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). Now answer the following questions The coordinate of the fourth vertex of the tetrahedron is :

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))

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)

Show that the lines (x-1)/(2)=(y-2)/(y-1)=(z-3)/(4) and (x-4)/(5)=(y-1)/(2)=z intersect

Using matrices, solve the following system of equations for x,y and z. 3x-2y+3z=8 , 2x+y-z=1 , 4x-3y+2z=4

Point of intersection of the line (x-1)/2 = (y-2)/3 = (z+3)/4 and the plane 2x+4y -z=1 is