Question

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

Answer 1

Heya...Empress Here!!

• Here are the steps!

• Draw a trapezium MNOP and note that lines MN//OP.

• Now from the center of MP, draw a line (XY) that extends to the center of NO and also draw a diagonal from M to O. Note that MO and XY intersect at point Z.

> Now to prove :- XY bisects MO and that is MZ = ZO.

• Proof :-

> In triangle MPO :-
• X is the midpoint of MP.
• XZ//OP

=> XZ bisects MO (Therom :- Converse of midpoint applied here).

= XY bisects MO.

• Thus, proven.

• Felicitations
- Kaileena

Answer 2

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.