Prove midpoints of the sides of a quadrilateral join to form a parallelogram.

Let \(ABCD\) be a quadrilateral with position vectors \(\mathbf{a}\), \(\mathbf{b}\), \(\mathbf{c}\), and \(\mathbf{d}\) for vertices \(A\), \(B\), \(C\), and \(D\) respectively. Let \(P\), \(Q\), \(R\), and \(S\) denote the midpoints of sides \(AB\), \(BC\), \(CD\), and \(DA\) respectively. Which statement correctly proves that \(PQRS\) is a parallelogram?

A quadrilateral \(PQRS\) has vertices at positions \(\mathbf{p}\), \(\mathbf{q}\), \(\mathbf{r}\), and \(\mathbf{s}\). The midpoints of sides \(PQ\), \(QR\), \(RS\), and \(SP\) are denoted by \(M\), \(N\), \(T\), and \(U\) respectively. (a) Express the position vectors of \(M\), \(N\), \(T\), and \(U\) in terms of \(\mathbf{p}\), \(\mathbf{q}\), \(\mathbf{r}\), and \(\mathbf{s}\). (1 mark) (b) Prove that \(MNTU\) is a parallelogram by showing that \(\vec{MN} = \vec{UT}\). (2 marks)