Question

find the coordinates of the centroid of a triangle formed by the three points (3,-4),(4,7) and (2,9)

Answer 1

Answer:

(x1+x2+x33, y1+y2+y33).

Step-by-step explanation:

Let A (x1, y1), B (x2, y2) and C (x3, y3) are  the three vertices of the ∆ABC .

Let D be the midpoint of side BC.

Since, the coordinates of B (x2, y2) and C (x3, y3), the coordinate of the point D are (x2+x32, y2+y32).

Let G(x, y) be the centroid of the triangle ABC.

Then, from the geometry, G is on the median AD and it divides AD in the ratio 2 : 1, that is AG : GD = 2 : 1.

 

Therefore, x = {2⋅(x2+x3)2+1⋅x12+1} = x1+x2+x33

y = {2⋅(y2+y3)2+1⋅y12+1} = y1+y2+y33

Therefore, the coordinate of the G are (x1+x2+x33, y1+y2+y33)

Hence, the centroid of a triangle whose vertices are (x1, y1), (x2, y2) and (x3, y3) has the coordinates (x1+x2+x33, y1+y2+y33).

Note: The centroid of a triangle divides each median in the ratio 2 : 1 (vertex to base).

Answer 2

Answer:

(3,4)

Step-by-step explanation:

Concept= Centroid

Given= The coordinates of triangle.

To find= The coordinates of centroid

Explanation=

We have been the question to find the coordinates of the centroid of a triangle formed by the three points (3,-4),(4,7) and (2,9).

So we know that when there is a ΔABC

The coordinates of A are taken as (x₁ , y₁)

B= (x₂ , y₂)

C= (x₃ , y₃)

Let the centroid point be (X,Y)

The centroid is the center of the triangle inside the triangle.

The formula for finding the Centroid is

X = (x₁ + x₂ + x₃)/3

Y= (y₁ + y₂ + y₃)/3

So we have been given the coordinates of a triangle in question, replacing them with our variables as,

(x₁ , y₁) = (3,-4)

(x₂ , y₂)=(4,7)

(x₃ , y₃)= (2,9)

Centroid is

X= ( 3+4+2)/3 = 9/3=3

Y= (-4 + 7 + 9)/3 = 12/3= 4

(X,Y) = (3,4)

Therefore the centroid of triangle formed is (3,4).

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