Question

If the centroid of the triangle formed by the points (a, b), (b, c) and (c, a) is at the origin, then a³ + b³ + c³ =

Options

• abc

• 0

• a + b + c

•  3 abc

Answer 1

The co-ordinates of the vertices are (a, b); (b, c) and (c, a)

The co-ordinate of the centroid is (0, 0)

We know that the co-ordinates of the centroid of a triangle whose vertices are `(x_1 ,y_1) ,(x_2 , y_2) ,(x_3 ,y_3)`  is

`((x_1 + x_2 + x_3 )/3 , ( y_1 + y_2 + y_3)/ 3)`

So,

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

Compare individual terms on both the sides-

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

Therefore,

`a + b+ c = 0`

We have to find the value of -

`a^3 + b^3 +c^3`

Now as we know that if,

a + b +c = 0

Then,

`a^3 + b^3 +c^3 =  3abc`