Mathematics
In a triangle ABC, AD is a median and E is mid-point of median AD. A line through B and E meets AC at point F.
Prove that : AC = 3AF.
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.
Draw DG || BF
In △ ADG,
⇒ DG || EF (Since, DG || BF)
Since, E is the mid-point of AD,
∴ F is the mid-point of AG (By converse of mid-point theorem)
∴ AF = FG ………(1)
In △ BCF,
⇒ DG || BF
As, AD is the median, D is the mid-point of BC.
∴ G is the mid-point of FC (By converse of mid-point theorem)
∴ FG = GC ………(2)
From equations (1) and (2), we get :
⇒ AF = FG = GC.
From figure,
⇒ AC = AF + FG + GC
⇒ AC = AF + AF + AF
⇒ AC = 3AF.
Hence, proved that AC = 3AF.
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