Question

Show that the line segment which joins the midpoints of the oblique sides of a trapezium is parallel to the parallel sides.

Answer 1

Let the trapezium be ABCD with E and F as the mid Points of AD and BC, Respectively Produce AD and BC to Meet at P.  

   

In Δ PAB, DC || AB.
Applying Thales’ theorem, we get 

`(PD)/(DA)=(PC)/(CB)` 

Now, E and F are the midpoints of AD and BC, respectively 

⇒ `(PD)/(2DE)=(PC)/(2CF)` 

⟹ `(PD)/(DE)=(PC)/(CF)` 

Applying the converse of Thales’ theorem in Δ PEF, we get that DC  

Hence, EF || AB.
Thus. EF is parallel to both AB and DC.
This completes the proof.