If the centroid of ΔABC having vertices A (a,b) , B (b,c) and C (c,a) is the origin, then find the value of (a+b+c).

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#### Solution

The given points are A (a,b) , B (b,c) and C (c,a)

Here,

`(x_1 = a , y_1=b),(x_2 = b, y_2 =c) and (x_3 = c, y_3= a)`

Let the centroid be (x , y) .

Then,

`x= 1/3(x_1 +x_2 +x_3)`

`= 1/3 (a+b+c)`

`=(a+b+c)/3`

`y= 1/3 (y_1+y_2+y_3)`

`=1/3 (b+c+a)`

`=(a+b+c)/3`

But it is given that the centroid of the triangle is the origin. Then, we have

`=(a+b+c)/3 = 0`

`⇒ a+b+c =0`

Concept: Area of a Triangle

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