Question

The centroid of a triangle divides each median in the ratio __________.

• A. $1:1$
• B. $2:3$
• C. $2:1$
• D. $3:1$

Answer 1

Answer: (C)

$2:1$

Given $G$ is centroid,

$AD,BE,CF$ are median.

To Prove,

$\frac{AG}{GD}=\frac{BG}{GE}=\frac{CG}{GF}=\frac{2}{1}$

Construction : Produce $AD$ to $K$ such that $AG=GK$, join $BK$ and $CK$

Proof : In $△ABK,$

$F$ and $G$ are mid points of $AB$ and $AK$ respectively

So, $FG\parallel BK$ [by the mid point therom]

Hence we can say that $GC\parallel BK$ ..... (A)

In $△AKC$

Similarly, $BG\parallel KC$ ..... (B)

By (A) and (B)

$BGCK$ is a parallelogram

In a parallelogram, diagonals bisect each other

So, $GD=DK$------(C)

$AG=GK$ [By construction]

$AG=GD+DK$

So, $AG=2GD$ [By (C)]

$\Rightarrow \frac{AG}{GD}=\frac{2}{1}$

Thus, the centroid of the triangle divides each of its median in the ratio $2:1$.