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Question
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).
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`
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