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

Correct option is (a) AE

We know that the centroid of a triangle is the point of intersection of the medians of a triangle and also centroid of a triangle divides each median in the ratio 2:1.

Then, we get,

\(\frac{AE}{OE} = \frac 21\) and \(\frac{CF}{OF} = \frac 21\) 

AE = 2OE           .........(1)

And CF = 2OF     .........(2)

We know that the diagonals of a parallelogram bisect each other.

Therefore, AO = OC

AE + OE = CF + OF

From eqⁿ(1) and eqⁿ(2), we get

2OE + OE = 2OF + OF

3OE = 3OF

OE = OF  .......(3)

Similarly we can prove,

AE = CF   ..........(4)

From eq(1), eq(2), eq(3) and eq(4), we get

AE = 2OE = 2OF = CF    .........(5)

Now, we have

OE + OF = OE + OE.      (from eq(3))

EF = 2OE

From equation (5), we get

EF = AE

Hence, this is the answer.