Question

If $ABCD$ is a parallelogram and $E$ and $F$ are the centroids of triangles $ABD$ and $BCD$ respectively, then $EF$ equals

• A. $AE$
• B. $BE$
• C. $CE$
• D. $DE$

Answer 1

The correct option is A $AE$

$ABCD$ is a parallelogram.

Let the diagonal of parallelogram meet at $O$.
Now, in $△ABD$
$OE=\frac{1}{3}AO$ (Centroid divides the median in the ratio $2:1$)
Also, In $△BCD$
$OF=\frac{1}{3}CO$ (Centroid divides the median in the ratio $2:1$)
Now, $OE+OF=\frac{1}{3}AO+\frac{1}{3}OC$
$EF=\frac{2}{3}AO$ (Since, diagonals bisect each other)
$EF=AE$ (Centroid divides the median in the ratio $2:1$)