Mathematics
In the adjoining figure, ABCD is a trapezium in which AB || DC and E is the mid-point of AD. If EF || AB meets BC at F, show that F is the mid-point of BC.
Mid-point Theorem
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Answer
Join AC. Let AC intersects EF at O.
Given,
AB || DC and EF || AB
∴ EF || AB || DC
By mid-point theorem,
The line segment joining the mid points of any two sides of a triangle is parallel to the third side and is equal to half of it.
By converse of mid-point theorem,
A line drawn through the midpoint of one side of a triangle, and parallel to another side, will bisect the third side.
Since,
⇒ EF || DC
⇒ EO || DC
In △ADC,
E is the mid-point of AD and EO || DC.
∴ O is the mid-point of AC. (By converse of mid-point theorem)
Given,
⇒ EF || AB
⇒ OF || AB
In △ABC,
O is the mid-point of AC and OF || AB.
∴ F is the mid-point of BC. (By converse of mid-point theorem)
Hence, proved that F is the mid-point of BC.
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