Mathematics
In triangle ABC, angle B is obtuse. D and E are mid-points of sides AB and BC respectively and F is a point on side AC such that EF is parallel to AB. Show that BEFD is a parallelogram.
Mid-point Theorem
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Answer
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,
The straight line drawn through the mid-point of one side of a triangle parallel to another, bisects the third side.
In △ ABC,
E is the mid-point of BC and FE || AB.
∴ F is the mid-point of AC. (By converse of mid-point theorem)
Since,
⇒ FE || AB
∴ FE || BD.
D and F are mid-point of sides AB and AC respectively.
∴ DF || BC (By mid-point theorem)
∴ DF || BE.
Since, opposite sides of quadrilateral BEFD are parallel.
Hence, proved that BEFD is a parallelogram.
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