$ \mathrm{ABCD} $ is a trapezium in which $ \mathrm{AB} \| \mathrm{DC}, \mathrm{BD} $ is a diagonal and $ \mathrm{E} $ is the mid-point of $ \mathrm{AD} $. A line is drawn through E parallel to $ \mathrm{AB} $ intersecting $ \mathrm{BC} $ at $ \mathrm{F} $ (see below figure). Show that $ \mathrm{F} $ is the mid-point of $ \mathrm{BC} $.
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Given:
\( \mathrm{ABCD} \) is a trapezium in which \( \mathrm{AB} \| \mathrm{DC}, \mathrm{BD} \) is a diagonal and \( \mathrm{E} \) is the mid-point of \( \mathrm{AD} \).
A line is drawn through E parallel to \( \mathrm{AB} \) intersecting \( \mathrm{BC} \) at \( \mathrm{F} \)
To do:
We have to show that \( \mathrm{F} \) is the mid-point of \( \mathrm{BC} \).
Solution:
Let $P$ be the point of intersection of $BD$ and $EF$.
In $\triangle ABD$,
$EP \| AB$
$E$ is the mid-point of $AD$.
We know that,
A line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side.
Therefore,
$P$ is the mid-point of $BD$.
In $\triangle BCD$,
$PF \| CD$
$P$ is the mid-point of $BD$.
By converse of mid-point theorem,
$F$ is the mid-point of $CB$.
Hence proved.
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