ABCD is a quadrilateral in which AD=BC. If P,Q,R,S be the mid-points of AB,AC,CD,BD respectively,show that PQRS is a rhombus.
join BD. consider triangle ABD.
P and S are the mid points of the sides of the triangle so line joining them is parallel to the third side BD and half of it...(mid pt thm)
QR is parallel to BD and half of it.
so PQRS is a parallelogram...9one pair of opp sides is paralel and equal)
in triangle APS,
in tringle BPQ,
PQ^2=PB^2+BQ^2.. Pythagoras thm
but AP=PB.. P is mid pt given
AS=BQ (since AD=BC)
Thus we have a parallelogram PQRS where one pair of adjacent sides is equal, hence it's a rhombus.