Prove that the volume of the tetrahedron and that formed by the centroids of the faces are in the ratio of `27:1.`

If V be the volume of a tetrahedron and V ' be the volume of another tetrahedran formed by the centroids of faces of the previous tetrahedron and V=K V^(prime),t h e nK is equal to a. 9 b. 12 c. 27 d. 81

If V be the volume of a tetrahedron and V ' be the volume of another tetrahedran formed by the centroids of faces of the previous tetrahedron and V=K V^(prime),t h e nK is equal to a. 9 b. 12 c. 27 d. 81

Find the ratio of the volume of tetrahedron with that of the tetrahedron formed by the centroids of its faces. Given Volume of tetrahedron =1/3 times area of base triangle times height of vertex.

Prove that the lines joining the vertices of a tetrahedron to the centroids of opposite faces are concurrent.

i. Show that the lines joining the vertices of a tetrahedron to the centroids of opposite faces are concurrent. ii. Show that the joins of the midpoints of the opposite edges of a tetrahedron intersect and bisect each other.

Prove that the formula for the volume V of a tetrahedron, in terms of the lengths of three coterminous edges and their mutul inclinations is V^2=(a^2b^2c^2)/36 |(1,cosphi,cospsi),(cosphi,1,costheta),(cospsi, costheta, 1)|

Prove that the centroid of any triangle is the same as the centroid of the triangle formed by joining the middle points of its sides

The four lines drawing from the vertices of any tetrahedron to the centroid to the centroid of the opposite faces meet in a point whose distance from each vertex is 'k' times the distance from each vertex to the opposite face, where k is

Find the equation of a plane passing through (1, 1, 1) and parallel to the lines L_1 and L_2 direction ratios (1, 0,-1) and (1,-1, 0) respectively. Find the volume of the tetrahedron formed by origin and the points where this plane intersects the coordinate axes.