The centroid of tetrahedron with vertices A (3,-5, x), B (5,4,2) , C (7,-7, y ), D (1,0, z ) is G (4,-2,2) , then x+ y + z =

Step-by-step solution

The centroid of a tetrahedron with vertices (x₁, y₁, z₁) , (x₂, y₂, z₂) , (x₃, y₃, z₃) , and (x₄, y₄, z₄) is located at G ( x₁+x₂+x₃+x₄ 4 , y₁+y₂+y₃+y₄ 4 , z₁+z₂+z₃+z₄ 4 ) . Given the vertices A(3,-5,x) , B(5,4,2) , C(7,-7,y) , D(1,0,z) and centroid G(4,-2,2) , we verify the x and y coordinates: 3+5+7+1 4 = 16 4 = 4 -5+4-7+0 4 = -8 4 = -2 Both match the given G . For the z -coordinate: x+2+y+z 4 = 2 x+y+z+2 = 8 x+y+z = 6 The correct option is B .