Question

If $A(2, -3)$ and $B(-2, 1)$ are two vertices of a triangle and third vertex moves on the line $2x + 3y = 9$, then the locus of the centroid of the triangle is:

• A. 2x + 3y = 3
• B. 2x - 3y = 1
• C. x - y = 1
• D. 2x + 3y = 1

Answer 1

Answer: (D)

2x + 3y = 1

Answer 2

Let the vertices of the triangle be $A(x_1, y_1) = (2, -3)$, $B(x_2, y_2) = (-2, 1)$, and $C(x_3, y_3)$. The centroid $G(x, y)$ is given by $x = \frac{x_1 + x_2 + x_3}{3}$ and $y = \frac{y_1 + y_2 + y_3}{3}$. Substituting the coordinates of A and B: $x = \frac{2 + (-2) + x_3}{3} = \frac{x_3}{3} \implies x_3 = 3x$ $y = \frac{-3 + 1 + y_3}{3} = \frac{-2 + y_3}{3} \implies y_3 = 3y + 2$ Since C moves on the line $2x_3 + 3y_3 = 9$, substitute the expressions for $x_3$ and $y_3$: $2(3x) + 3(3y + 2) = 9$ $6x + 9y + 6 = 9$ $6x + 9y = 3$ Divide by 3: $2x + 3y = 1$