Question

E and F are respectively the mid points of non-parallel sides AD and BC of a trapezium ABCD. Prove that EF is parallel to AB and EF = 1/2(AB +CD).​

Answer 1

Answer:

Solution

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Given: A trapezium ABCD in which E and F are respectively the mid-points of the non-parallel sides AD and BC

To prove: EF∣∣AB and EF=

2

1

(AB+CD)

Construction: Join DF and produce it to intersect AB produced at G.

Proof: In ΔCFD and ΔBFG, we have

DC∣∣AB

∴∠C=∠3 [ALternate interior angles]

CF=BF

∠1=∠2 [Vertically opposite angles]

So, by ASA criterion of congruence, we have

ΔCFD≅ΔBFG

∴CD=BG(CPCT)

In ΔDAG,EF joins mid-points of sides AD and GD respectively

∴EF∣∣AG [∵ Mid-point theorem]

⇒EF∣∣AB

So, EF=

2

1

AG[ Mid-point theorem]

EF=

2

1

(AB+AG)

EF=

2

1

(AB+CD)[∵CD=BG]

Hence, proved.