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Midpoints of any quadrilateral form a parallelogram

Midpoints of any quadrilateral form a parallelogram

IP 4
8 marks
  1. In triangle \(OAB\), \(M\) and \(N\) are the midpoints of \(OA\) and \(OB\). Using vectors, show that \(MN\) is parallel to \(AB\) and half its length.[3]
  2. \(ABCD\) is any quadrilateral, and \(P\), \(Q\), \(R\), \(S\) are the midpoints of \(AB\), \(BC\), \(CD\), \(DA\). Using position vectors \(\mathbf a\), \(\mathbf b\), \(\mathbf c\), \(\mathbf d\), prove that \(PQRS\) is a parallelogram.[4]
  3. State the condition on the diagonals of \(ABCD\) for \(PQRS\) to be a rhombus.[1]

Solution:

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Answer:(a) \(\overrightarrow{MN}=\dfrac12\overrightarrow{AB}\) (b) \(\overrightarrow{PQ}=\overrightarrow{SR}=\dfrac12\overrightarrow{AC}\) (c) \(AC=BD\)

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