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Diagonals that bisect each other

Diagonals that bisect each other

IP 4
8 marks
  1. \(OABC\) is a parallelogram with \(\overrightarrow{OA}=\mathbf a\) and \(\overrightarrow{OC}=\mathbf c\). Show that the midpoints of the diagonals \(OB\) and \(AC\) are the same point.[3]
  2. Conversely, \(OABC\) is a quadrilateral whose diagonals \(OB\) and \(AC\) bisect each other. Using position vectors \(\mathbf a\), \(\mathbf b\), \(\mathbf c\) of \(A\), \(B\), \(C\) relative to \(O\), prove that \(OABC\) is a parallelogram.[3]
  3. Points \(P(1,1)\), \(Q(5,2)\) and \(R(6,6)\) are three vertices of a parallelogram \(PQRS\). Use the result of (b) to find \(S\).[2]

Solution:

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Answer:(a) both midpoints are \(\dfrac12(\mathbf a+\mathbf c)\) (b) \(\mathbf b=\mathbf a+\mathbf c\) gives \(\overrightarrow{AB}=\overrightarrow{OC}\) (c) \(S=(2,5)\)

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