If a line is parallel to the base of a trapazium and bisects one of the non-parallel sides, then prove that it bisects either diagonal of the trapazium.

Given: ABCD is a trapezium. EF is parallel to DC. and it bisects one of the non parallel sides.

TPT: EF bisects either of the diagonals of the trapezium

proof:

let EF bisects AD, i.e. E is the mid-point of AD.

now in the triangle ADC,

E is the mid-point of AD and EM || DC (as EF is parallel to DC)

From the converse of the mid point theorem: the straight line drawn through the mid-point of one side of a triangle parallel to another , bisects the third side.

Therefore M is the mid point of AC.

Thus if EF bisects AD at E , it also bisects the diagonal AC.

Similarly we can show that if EF bisects BD at F , it also bisects the diagonal BD.