Question

In a trapezium ABCD,AB and CD are parralell to each other,E ia the midpoint of AD amd F is the mid point of BC.prove that AB+DC=2EF

Answer 1

Given that , ABCD is a trapezium, where AB parallel to DC.E is the midpoint of AD.
EF  parallel to AB. We had to prove that AB+CD=2 EF.
Construction: join AC.
Proof: EF  parallel to AB but AB parallel to DC.therefore, EF parallel to DC. 
           In tri ADC ,E is the midpoint of side AD and EO parallel to DC.
           therefore O is the midpoint of AC , by converse of Mid point theorem.
           Now ,E and O are midpoints of sides AD and AC of tri ADC . therefore EO = 1/2 CD ..(1), by MPT.
          similarly , OF = 1/2 AB.....(2)
           (1)+(2) implies
           EO +OF =1/2 CD+1/2 AB
           EF =1/2[AB + CD]
therefore, AB + CD =2 EF
          Thus proved

that's all