Mathematics - 3D-Coordinate Geometry Question with Solution | TestHub
(i) Consider the four vertices and six edge midpoints of a tetrahedron. Fix one vertex P1. How many groups of four distinct points containing P1 are coplanar? (ii) For four non-coplanar points A,B,C,D, how many planes are equidistant from all four?
Solution:
(i) Each of the three faces through the fixed vertex contains six marked points. Choose any three other points on it. The only extra groups consist of the fixed vertex, another vertex, their midpoint and the opposite-edge midpoint. There are three such groups. An affine coordinate check with vertices at the origin and twice the three coordinate unit vectors gives no others.
(ii) Assign each vertex signed distance plus or minus r. Reversing all signs gives the same plane. The all-equal assignment has zero normal and is not a plane; every other assignment determines one plane.
