Question

ABCD is a trapezium in which side AB is parallel to DC and E is the mid point of side AD. If F is the mid point of BC,such that line segments EF is parallel to DC,then prove that EF=1/2(AB+DC)

Answer 1

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Answer 2

SOLUTION:

Given: A trapezium ABCD in which AB || DC, E is the mid-point of AD and F is a point on BC such that EF || DC.

To Prove: EF = 1/2(AB + DC)

Proof: In △ADC, E is the mid-point of AD and ED || DC ______[Given]

∴ G is the mid-point of AC ____[∵ Segment joining the mid-points of the two sides of a △is half of the third side]

∴ EG = 1/2DC ________(i)

Now, ABCD is a trapezium in which AB || DC.

But, EF || DC

∴ EF || AB

→ GF || AB

In △ABC, G is the mid-point of AC(proved above) and GF || AB.

∴ F is the mid-point of BC _____[∵ Segment joining the mid-points of the two sides of a △is half of the third side]

∴ GF = 1/2AB __________(ii)

From (i) and (ii), we have

→ EG + GF = 1/2DC + 1/2AB

→ EF = 1/2(AB + DC) ________[PROVED]