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 =

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The centroid of a tetrahedron with vertices $(x_{1}, y_{1}, z_{1})$ , $(x_{2}, y_{2}, z_{2})$ , $(x_{3}, y_{3}, z_{3})$ , and $(x_{4}, y_{4}, z_{4})$ is given by $(\frac{x _{1} + x _{2} + x _{3} + x _{4}}{4}, \frac{y _{1} + y _{2} + y _{3} + y _{4}}{4}, \frac{z _{1} + z _{2} + z _{3} + z _{4}}{4})$ .

Given vertices are $A (3, - 5, x)$ , $B (5, 4, 2)$ , $C (7, - 7, y)$ , $D (1, 0, z)$ and centroid is $G (4, - 2, 2)$ .

Equating the z-coordinates, we get:

$\frac{x + 2 + y + z}{4} = 2$

$x + y + z + 2 = 8$

$x + y + z = 6$

Answer: 6

Mathematics - 3D Coordinate Geometry Question with Solution | TestHub

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