A tetrahedron has vertices with position vectors a,b,c,d. Let M and N be the midpoints of AB and CD, respectively. Prove that the midpoint of MN has position vector 4a+b+c+d, and hence show that the three line segments joining the midpoints of opposite edges of a tetrahedron bisect one another.
The midpoint vectors are m=(a+b)/2 and n=(c+d)/2. Hence the midpoint of MN is 2m+n=4a+b+c+d.
Pairing the opposite edges as (AC,BD) gives the same sum divided by 4, as does (AD,BC). All three segments therefore share a midpoint and bisect one another.
Determines the vectors of M and N.
[2 marks]
Determines the midpoint of MN.
[2 marks]
Applies the result to the other two pairings.
[1 mark]
Makes the geometric conclusion.
[1 mark]
Practice question aligned to the current QCAA syllabus; review the worked solution and marking criteria.