Mathematics
D and F are mid-points of sides AB and AC of a triangle ABC. A line through F and parallel to AB meets BC at point E.
(i) Prove that BDFE is a parallelogram.
(ii) Find AB, if EF = 4.8 cm
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.
(i) In △ ABC,
Since, D and F are mid-points of sides AB and AC.
∴ DF || BC (By mid-point theorem)
Given,
⇒ FE || AB
∴ FE || BD.
Since, opposite sides of quadrilateral BDFE are parallel.
Hence, proved that BDFE os a parallelogram.
(ii) In △ ABC,
⇒ FE || AB and F is mid-point of AC.
∴ E is mid-point of BC (By converse of mid-point theorem)
∴ EF = (By mid-point theorem)
⇒ AB = 2EF = 2 × 4.8 = 9.6 cm.
Hence, AB = 9.6 cm.
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