Question

If the centroid of the triangle formed by the point \[(a,b),(b,c)\,and\,(c,a)\] is at the origin then find the value of \[({{a}^{3}}+{{b}^{3}}+{{c}^{3}})\] .
(A). \[abc\]
(B). \[a+b+c\]
(C). \[3abc\]
(D). 0

Answer 1

Hint: Assume a \[\Delta ABC\] having the coordinates of the vertex A, B, and C as \[(a,b),(b,c)\,and\,(c,a)\] respectively. We know the formula of the x-coordinate of the centroid of the triangle, the x coordinate of the centroid of the triangle = \[\left( \dfrac{\text{sum}\,\text{of x-coordinates of all the vertex of the triangle}}{3} \right)\] . It is given that the centroid of the triangle is at origin. Now, solve it further.

Complete step-by-step solution -
According to the question, we have a triangle and the coordinates of the vertex of the triangle are
\[(a,b),(b,c)\,and\,(c,a)\] . It is also given that the centroid of this triangle is at origin. We know the coordinate of the origin.

The Coordinate of vertex A = \[(a,b)\] ……………….(1)
The Coordinate of vertex A = \[(b,c)\] ……………….(2)
The Coordinate of vertex A = \[(c,a)\] ……………….(3)
The x-coordinate of the centroid of the triangle = 0 ……………………….(4)
The y-coordinate of the centroid of the triangle = 0 ……………………….(5)
We know the formula to find the coordinate of the centroid of the triangle,
The x coordinate of the centroid of the triangle = \[\left( \dfrac{\text{sum}\,\text{of x-coordinates of all the vertex of the triangle}}{3} \right)\] ………………………(6)
The y coordinate of the centroid of the triangle = \[\left( \dfrac{\text{sum}\,\text{of y-coordinates of all the vertex of the triangle}}{3} \right)\] ………………………(7)
Now, from equation (1), equation (2), equation (3) and equation (6), we get
The x coordinate of the centroid of the triangle = \[\left( \dfrac{a+b+c}{3} \right)\] ………………………….(8)
But from equation (4), we have the x coordinate of the centroid of the triangle.
On, comparing equation (4) and equation (8), we get
\[\left( \dfrac{a+b+c}{3} \right)=0\]
\[\Rightarrow a+b+c=0\] ………………………..(9)
We know the property that, if \[(a+b+c=0)\] then \[({{a}^{3}}+{{b}^{3}}+{{c}^{3}})\] is equal to \[3abc\].
From equation (9), we have \[(a+b+c=0)\] . So, using the property we can say that \[({{a}^{3}}+{{b}^{3}}+{{c}^{3}})\] is equal to \[3abc\].
Hence, the correct option is option (C).

Note: In this question, one might get confused because we have not used the information provided for the y coordinates of the centroid. But if we use the y coordinates then on comparing, we will again get the same equation.
The Coordinate of vertex A = \[(a,b)\]
The Coordinate of vertex B = \[(b,c)\]
The Coordinate of vertex C = \[(c,a)\]
The y coordinate of the centroid of the triangle = \[\left( \dfrac{b+c+a}{3} \right)\] ………………….(1)
It is given that the centroid of the triangle is at origin. We know the coordinate of the origin which is \[(0,0)\] .
The y-coordinate of the centroid of the triangle = 0 ……………………….(2)
On comparing equation (1) and equation (2), we get
\[\left( \dfrac{b+c+a}{3} \right)=0\] .
We can see that we got the same equation.