Question

Two vertices of a triangle \( ABC \) are \( A(2, 3) \) and \( B(1, -3) \). If the centroid is \( (3, 0) \), then the coordinates of the third vertex \( C \) are:

• A. \( (5, 2) \)
• B. \( (1, 3) \)
• C. \( (6, 0) \)
• D. \( (2, -3) \)

Hint:

The centroid of a triangle is the average of the coordinates of its vertices.

Answer 1

The Correct Option is A

The coordinates of the centroid \( G \) of a triangle are given by: \[ G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right), \] where \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \) are the vertices of the triangle. Given \( G(3, 0) \), \( A(2, 3) \), and \( B(1, -3) \), we use the centroid formula: \[ \left( 3, 0 \right) = \left( \frac{2 + 1 + x_3}{3}, \frac{3 + (-3) + y_3}{3} \right). \] From the x-coordinate: \[ \frac{2 + 1 + x_3}{3} = 3 \quad \Rightarrow \quad 3 + x_3 = 9 \quad \Rightarrow \quad x_3 = 6. \] From the y-coordinate: \[ \frac{3 + (-3) + y_3}{3} = 0 \quad \Rightarrow \quad y_3 = 2. \] Thus, the coordinates of \( C \) are \( (6, 2) \). Thus, the correct answer is \( \boxed{(5, 2)} \).