Question

in a triangle abc d is the midpoint of bc and e is the midpoint of ad. if be is produced meets ac in f, then prove that af=1/3ac

Answer 1

in triangle ABC,ADis the median .

in triangle ABC,ADis the median . E is themid point of AD. BE when produce meet ac at F. show that AF=1/3AC.

Given AD is the median of ΔABC.
E is the midpoint of AD, also BE meets AD at F after producing BE
Draw DG||BF
In ΔADG, E is the midpoint of AD and EF||DG.
By converse of midpoint theorem, we have
F is the midpoint of AG
That is AF = FG --- (1)
Similarly, in ΔBCF, D is the midpoint of BC and DG||BF
Therefore, G is the midpoint of CF
Hence FG = GC ---(2)
From equations (1) and (2), we get
AF = FG = GC --- (3)
From the figure we have, AF + FG + GC = AC
⇒ AF + AF + AF = AC [From (3)]
3AF = AC
Hence AF = (1/3) AC