Question

if a line is parallel to base of trapezium and bisect one of the non parallel sides, then prove that it bisect either diagonal of the trapezium

plz answer with fig.

Answer 1

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 PARALLEL DC (because e.g. parallel 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.

hope this helps you.