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ABCD is a parallelogram such that point E is the midpoint of side BC. If DE and AB when produced meet at F, then which of the following is true?

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AF=2 AB

AB = 2 AF

AB = AF

No relation between AB and AF

answer is A.

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Detailed Solution

Given that, E is the midpoint of side BC in the parallelogram ABCD.
Let us look at the following diagram.
Question Image

Here, BE = CE ...(given)
AB || DC

∠ D C E = ∠ E B F

… (alternate angles with transversal DF)
Consider

Δ D C E

and

Δ B F E

∠ D C E = ∠ E B F

... (alternate angles with transversal DF)
BE = CE ...(given)

∠ D E C = ∠ B E F

... (vertically opposite angles)
Hence, by ASA rule

Δ D C E ≅ Δ B F E

.
DC = BF ...by CPCT
In parallelogram ABCD,
DC = AB = BF.

A F = A B + B F A F = A B + A B A F = 2 A B

If DE and AB when produced meet at F, then AF=2 AB.
Option 1 is the correct choice.

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