Question

If the centroid of the triangle formed with \((a, b), (b, c)\) and \((c, a)\) is \(O(0, 0),\) then \(a^3 + b^3 + c^3 =\)

• A. abc
• B. 2abc
• C. -3abc
• D. 3abc

Answer 1

The Correct Option is D

Correct answer: 3abc 

Explanation:
Let the vertices of the triangle be: 
\( A(a, b),\ B(b, c),\ C(c, a) \) 
The centroid \( O \) of a triangle with vertices \( (x_1, y_1), (x_2, y_2), (x_3, y_3) \) is given by: \[ \left( \frac{x_1 + x_2 + x_3}{3},\ \frac{y_1 + y_2 + y_3}{3} \right) \] Given that the centroid is \( O(0, 0) \), we equate: \[ \frac{a + b + c}{3} = 0 \Rightarrow a + b + c = 0 \] Cube both sides: \[ (a + b + c)^3 = 0 \] Use identity: \[ a^3 + b^3 + c^3 + 3(a + b)(b + c)(c + a) = 0 \] But we will use a simpler identity when \( a + b + c = 0 \): \[ a^3 + b^3 + c^3 = 3abc \]

Hence, \( a^3 + b^3 + c^3 = {3abc} \)