InterviewSolution
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InterviewSolution
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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. |
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Answer» Solution :i. `G_(1)`, the centroid of `DeltaBCD`, is `(vecb+vecc+vecd)/(3) and A` is `VECA`. The position vector of point G which divides `AG_(1)` in the ratio 3 : 1 is `""(3*(vecb+vecc+vecd)/(3)+1*veca)/(3+1)=(veca+vecb+vecc+vecd)/(4)` The symmetry of the result shows that this point will also lie on `BG_(2)`, `CG_(3) and DG_(4)` (where `G_(2), G_(3), G_(4)` are centroids of faces ACD, ABD and ABC, respectively). Hence, these FOUR lines are concurrent at point `(veca+vecb+vecc+vecd)/(4)` , which is called the centroid of the tetrahedron. ii. The midpoint of `DA` is `(veca+vecd)/(2)`and the of `BC` is `(vecb+vecc)/(2)` and the midpoint of these midpoint is `(veca+vecb+vecc+vecd)/(4)` and symmetry of the result proves the fact. 
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