Question

Choose the most appropriate options.
If \( A(2,3) \) and \( B(-2,1) \) are two vertices of a triangle and the third vertex moves on the line \( 2x + 3y = 9 \), then the locus of the centroid of the new set of observations will be the triangle is

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

Hint:

The centroid of a triangle lies at the average of the coordinates of its three vertices.

Answer 1

The Correct Option is A

The centroid \( G \) of a triangle with vertices \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \) has coordinates: \[ G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \] Given that \( A(2, 3) \) and \( B(-2, 1) \), and the third vertex moves along the line \( 2x + 3y = 9 \), the locus of the centroid will follow the equation \( 2x + 3y = 3 \).