In a trapezium ABCD, E and F be the midpoints of the diagonals AC and BD respectively. Then, EF = ?
Let E and F are midpoints of the diagonals AC and BD of trapezium ABCD respectively.
Draw DE and produce it to meet AB at G.
Consider DAEG and DCED
∠AEG = ∠CED (vertically opposite angles)
AE = EC (E is the midpoint of AC)
∠ECD = ∠EAG (alternate angles)
ΔAEG ≅ ΔCED
⇒ DE = EG → (1)
And AG = CD → (2)
E is the midpoint of DG [From (1)]
F is the midpoint of BD
∴ EF is parallel to GB
⇒ EF is parallel to AB
⇒ EF is parallel to AB and CD
Also, EF = 12 GB
⇒EF = 12 (AB − AG)
⇒ EF = 12 (AB − CD) [From (2)]