Midpoints of the sides of a quadrilateral

\(ABCD\) is any quadrilateral with vertices \(A(a_1,a_2)\), \(B(b_1,b_2)\), \(C(c_1,c_2)\) and \(D(d_1,d_2)\). \(P\), \(Q\), \(R\) and \(S\) are the midpoints of \(AB\), \(BC\), \(CD\) and \(DA\) respectively.

1. Show that the diagonals \(PR\) and \(QS\) of \(PQRS\) have the same midpoint, and deduce that \(PQRS\) is a parallelogram.[3]
2. Show that \(PQ=\dfrac{1}{2}AC\).[2]
3. For the quadrilateral with \(A(0,0)\), \(B(8,0)\), \(C(8,6)\) and \(D(k,6)\), find the value of \(k\) for which \(PQRS\) is a rhombus, explaining why any other value is rejected.[3]