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Question
Question 4
ABCD is a trapezium in which AB || 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). Show that F is the mid - point of BC.
ABCD is a trapezium in which AB || 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). Show that F is the mid - point of BC.
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Solution
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 bisects the third side. In ΔABD,
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.
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 bisects the third side. In ΔABD,
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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