Question

In any triangle, the centroid divides the median in the ratio:

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

Answer 1

Answer: (B)

$2:1$

In the above figure, the two triangles $AEF$ and $ACB$ are similar as $\overline{AC}=2\overline{AE},\overline{AB}=2\overline{AF}$ and $\angle A$ is common for both triangles $AEF$ and $ACB$.

It follows that as triangles $AEF$ and $ACB$ are similar, the lines $EF$ and $CB$ are parallel and $CB=2\times EF$.

Now consider the triangles $EGF$ and $BGC$. Because the lines $EF$ and $CB$ are parallel, therefore,

$\angle GEF=\angle GBC$ and $\angle EFG=\angle GCB$

Also, as they are opposite angles.

Thus, triangles $EGF$ and $BGC$ are similar. Furthermore, because $CB=2\times EF$, this means that the length of the sides of the triangles $BGC$ and $EGF$ are in the ratio $2:1$ and we have

$BG:GE=CG:GF=2:1$

Similarly, by constructing a line from point $D$ to $F$, and using the two similar triangles $DFG$ and $AGC$, we can prove that

$AG:GD=2:1$

Therefore, the centroid divides the median in the ratio $2:1$.

Hence, option B is correct.