prove that the line segment joining the midpoints of the diagonals of a Trapeziu...

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September 2021 2 18 Report

prove that the line segment joining the midpoints of the diagonals of a Trapezium is parallel to each of the parallel sides and is equal to half of the difference of these sides

mubaraka83 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 midpoint of AC)

∠ECD = ∠�EAG (alternate angles)

ΔAEG ≅ ΔCED

⇒ DE = EG → (1)

And AG = CD → (2)

In ΔDGB

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 = ½ GB

⇒EF = ½ (AB − AG) ⇒ EF = ½ (AB − CD) [From (2)]

mubaraka83 so say in what u want i will help u

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