ABCD is a trapezium in which AB $\parallel$ DC, BD is a diagonal and E is the mid - point of AD. A line is drawn through E parallel to AB intersecting BC at F ( See the given figure). Mark the correct answer option.

E F = F C

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C B = 2 C F

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C F = F B

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A B^{2} + B C^{2} = B D^{2}

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Solution

The correct options are
B $C B = 2 C F$
C $C F = F B$
Let EF intersect DB at G.

By converse of mid-point theorem, we know that a line drawn through the mid-point of any side of a triangle and parallel to another side, bisect the third side.
In $Δ$ ABD, EF || AB and E is the mid-point of AD.
Therefore, G will be the mid-point of DB.
As EF || AB and AB || CD, $∴$ EF || CD ( two lines parallel to the same line are parallel to each other).
In $Δ$ BCD, GF || CD and G is the mid-point of line BD. Therefore, by using converse of midpoint theorem, F is the mid-point of BC.

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ABCD is a trapezium in which AB \(\parallel\) DC, BD is a diagonal and E is the mid - point of AD. A line is drawn through E parallel to AB intersecting BC at F ( See the given figure). Mark the correct answer option.