Name: Circumcentre of a Triangle
Uploaded: 2019-07-19
Duration: 7 min 3 s
Description: The circumcentre of a triangle is one of the important topics in Geometry. Learn about this concept along with examples by watching this video.

Question

What is centroid? Prove that centroid of any triangle divides its median in the ratio

2 : 1

from the vertex.

Accepted Answer

Centroid is a point at which three medians of any triangle intersect each other.

Given, Let $G$ be the centroid of $∆ A B C$ , i.e., three medians $A D, B E$ and $C F$ of the $∆ A B C$ meet at a point $G$ .

We need to prove, $A G : G D = B G : G E = C G : G E = 2 : 1$ .

Construction: Let us extend $A G$ to $H$ such that $A G = G H$ . Join $B, H$ and $C, H$ .

Proof: In $∆ A B H$ , $F$ is the mid-point of $A B$ and $G$ is the mid-point of $A H$ .

$∴ F G ∥ G H$ or $G C ∥ B H . . . . . . . . . . . . . . (i)$

Now, in $∆ A C H$ , $E$ is the mid-point and $G$ is the mid-point of $A H$ .

$∴ G E ∥ H C$ or $B G ∥ H C . . . . . . . . . . . . . . (ii)$

From $(i)$ and $(ii)$ , we get, in a quadrilateral $B C G H$ , $G C ∥ B H$ and $B G ∥ H C$ .

$∴ B G C H$ is a parallelogram.

$\Rightarrow$ Diagonals $B C$ and $G H$ bisect each other. i.e., $G D = H D$ .

$∴ G D = \frac{1}{2} G H$ or $G D = \frac{1}{2} A G$

$\Rightarrow \frac{A G}{G D} = \frac{2}{1}$ or $A G : G D = 2 : 1$

Similarly, it can be proved that $B G : G E = 2 : 1$ and $C G : G E = 2 : 1$ .

Hence, proved.

What is centroid? Prove that centroid of any triangle divides its median in the ratio $2 : 1$ from the vertex.

Important Questions on Theorems on Concurrence

Learn and Prepare for any exam you want !

What is centroid? Prove that centroid of any triangle divides its median in the ratio 2:1 from the vertex.

Important Questions on Theorems on Concurrence

Learn and Prepare for any exam you want !

What is centroid? Prove that centroid of any triangle divides its median in the ratio $2 : 1$ from the vertex.