Question

Find the coordinates of the centroid of the triangle whose vertices are x1 y1 x2 y2 and x3 y3

Answer 1

Answer:

Step-by-step explanation:

Let A(x1,y1), B(x2,y2) and C(x3,y3) be the vertices of a triangle ABC whose medians are AD,BE and CF respectively. So, D,E and F are respectively the mid points of

BC,CA and AB.

Coordinates of D are

Coordinates of a point G(x, y) dividing AD in the ratio 2 : 1 are

Answer 2

Coordinates of the centroid of the triangle is ( x₁  + x₂ + x₃)/3 ,  ( y₁  + y₂ + y₃)/3  whose vertices are (x₁ , y₁) ,  (x₂ , y₂) and (x₃ , y₃)  

Solution:

• The centroid of the triangle is the point of concurrency of the medians in the triangle.
• The segment connecting the vertex of a triangle and the midpoint of the opposite side is called the median of that triangle.
• For each vertex of a triangle, the centroid is two-thirds of the distance from the vertex to the midpoint of the opposite side.
• or centroid divided median in 2 : 1 ratio from the vertex

Step 1:

Assume that ΔPQR

P = (x₁ , y₁)

Q = (x₂ , y₂)

R = (x₃ , y₃)

Step 2:

Assume that RS is one of the Median hence S is the mid point of PQ

Coordinate of S = ( x₁  + x₂)/2 , ( y₁  + y₂)/2

Step 3:

Assume that G is  centroid then G will divide RS in 2 : 1 ratio

Coordinate of G = {2( x₁  + x₂)/2 + 1(x₃) } /(2 + 1)  ,  {2( y₁  + y₂)/2 + 1(y₃) } /(2 + 1)

Coordinate of G = ( x₁  + x₂ + x₃)/3 ,  ( y₁  + y₂ + y₃)/3

Coordinates of the centroid of the triangle is ( x₁  + x₂ + x₃)/3 ,  ( y₁  + y₂ + y₃)/3  whose vertices are (x₁ , y₁) ,  (x₂ , y₂) and (x₃ , y₃)