If the centroid of the triangle formed by the points (a,b),(b,c) and (c,a) is at the origin then$$a^3+b^3+c^3$$​ is equal to ….

Correct option is C

1. Given:

- The vertices of the triangle are (a, b), (b, c), (c, a) .

- The centroid of the triangle is at the origin (0, 0) .

2. Formula Used:

- The coordinates of the centroid of a triangle with vertices $$(x_1, y_1), (x_2, y_2), (x_3, y_3)$$​ are:

​$$\left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right)$$​

- If the centroid is at the origin, then: $$\frac{a + b + c}{3} = 0$$​

3. Solution:

Using the centroid formula:

​$$\frac{a + b + c}{3} = 0$$​

This implies: a + b + c = 0

To find $$a^3 + b^3 + c^3$$​ , use the identity:

​$$a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)$$​

Since a + b + c = 0 :

​$$a^3 + b^3 + c^3 - 3abc = 0$$​

Hence: $$a^3 + b^3 + c^3 = 3abc$$​

4. Answer:

The value of $$a^3 + b^3 + c^3$$​ is **3abc**.