Question

In $\Delta ABC$, D, E and F are mid-points of sides AB, BC and AC respectively.

• A. AE and DF bisect each other.
• B. AE and BF bisect each other.
• C. AC and DF bisect each other.
• D. AB and BF bisect each other.

Answer 1

The correct option is A AE and DF bisect each other.

Let G be the centroid of triangle ABC. Given E and F are the mid points of BC and AC respectively. Thus, by mid point theorem, $AD\parallel EF$
$AB=2EF$
$AD=EF$ (I) (D is mid point of AB)
Now, In $△ADG$ and $△GEF$,
$\angle AGD=\angle EGF$ (Vertically opposite angles)
$AD=EF$ (From I)
$\angle ADG=\angle GFE$ (Alternate angles for parallle lines EF and AD)
$△ADG\cong △EGF$ (ASA rule)
Thus, $AG=GE$ (Corresponding sides)
Also, $DG=GF$ (Corresponding sides)
Thus, AE and DF bisect each other at G.