Find the centroid of triangle if vertices are (2,1) (3,7)( 8,4 )

1.	Find the centroid of triangle if vertices are (2,1) (3,7)( 8,4 )
Answer» Given : - The vertices of the triangle are (2,1) , (3,7) & (8,4) Required to find : - Centroid of a triangle Concept used : - Before ! solving this question we need to know an important concept related to the centroid of a triangle . How to find the co-ordinates of the centroid of a triangle . Consider a traingle ABC such that the co-ordinates of the vertices of the triangle A = (X1,y1) , B = (x2,y2) & C = (X3,y3) . A centroid is a point where all the 3 ALTITUDES bisect each other . Since, the vertices of a triangle are 3 . so, the no. of altitudes are also 3 . A centroid divides the altitude of a triangle in the ratio of 1 : 2 . If we want to find the centroid of a triangle then we need to draw the corresponding altitudes . where all the 3 altitudes bisect that point is termed as the centroid . According to physics , it will be called as the centre of gravity . So, The formula to find the co-ordinates of centroid of a triangle is ; $\sf{ G = \left( \dfrac{ x_1 + x_2 + x_3}{ 3 } , \dfrac{ y_1 + y_2 + y_3 }{ 3} \right) }$ Here, G is the centroid of the traingle . USING this concept let's crack that question . Solution : - The vertices of the triangle are (2,1) , (3,7) & (8,4) . We need to find the co-ordinates of the centroid . So, Let the vertices of the traingle be ; A = (2,1) B = (3,7) C = (8,4) This implies ; x1 = 2 , y1 = 1 x2 = 3 , y2 = 7 x3 = 8 , y3 = 4 Now, Using the formula let's find the co-ordinates of the centroid . $\sf{ G = \left( \dfrac{ x_1 + x_2 + x_3}{ 3 } , \dfrac{ y_1 + y_2 + y_3 }{ 3} \right) }$ Substituting the values in it $\sf \longrightarrow G =\left( \dfrac{ 2 + 3 + 8}{3} , \dfrac{1 + 7 + 4}{3} \right) \\ \\ \sf \longrightarrow G = \left( \dfrac{13}{3} , \dfrac{12}{3}\right) \\ \\ \sf \longrightarrow G = \left( \dfrac{13}{3} , 4 \right) \\ \\ \\ \therefore \tt Co-ordinates \; of \; the \; centroid = \left( \dfrac{13}{3} , 4 \right)$

Answer»

Given : -

The vertices of the triangle are (2,1) , (3,7) & (8,4)

Required to find : -

Centroid of a triangle

Concept used : -

Before ! solving this question we need to know an important concept related to the centroid of a triangle .

How to find the co-ordinates of the centroid of a triangle .

Consider a traingle ABC such that the co-ordinates of the vertices of the triangle A = (X1,y1) , B = (x2,y2) & C = (X3,y3) .

A centroid is a point where all the 3 ALTITUDES bisect each other .

Since, the vertices of a triangle are 3 . so, the no. of altitudes are also 3 .

A centroid divides the altitude of a triangle in the ratio of 1 : 2 .

If we want to find the centroid of a triangle then we need to draw the corresponding altitudes . where all the 3 altitudes bisect that point is termed as the centroid .

According to physics , it will be called as the centre of gravity .

So,

The formula to find the co-ordinates of centroid of a triangle is ;

$\sf{ G = \left( \dfrac{ x_1 + x_2 + x_3}{ 3 } , \dfrac{ y_1 + y_2 + y_3 }{ 3} \right) }$

Here, G is the centroid of the traingle .

USING this concept let's crack that question .

Solution : -

The vertices of the triangle are (2,1) , (3,7) & (8,4) .

We need to find the co-ordinates of the centroid .

So,

Let the vertices of the traingle be ;

A = (2,1)

B = (3,7)

C = (8,4)

This implies ;

x1 = 2 , y1 = 1

x2 = 3 , y2 = 7

x3 = 8 , y3 = 4

Now,

Using the formula let's find the co-ordinates of the centroid .

$\sf{ G = \left( \dfrac{ x_1 + x_2 + x_3}{ 3 } , \dfrac{ y_1 + y_2 + y_3 }{ 3} \right) }$

Substituting the values in it

$\sf \longrightarrow G =\left( \dfrac{ 2 + 3 + 8}{3} , \dfrac{1 + 7 + 4}{3} \right) \\ \\ \sf \longrightarrow G = \left( \dfrac{13}{3} , \dfrac{12}{3}\right) \\ \\ \sf \longrightarrow G = \left( \dfrac{13}{3} , 4 \right) \\ \\ \\ \therefore \tt Co-ordinates \; of \; the \; centroid = \left( \dfrac{13}{3} , 4 \right)$