Question

The centroid of a triangle is (4, 1) and two vertices are (2, 3) and (7, 6) then the third vertex is

• A. (3, 6)
• B. (-3, 6)
• C. (-3, -6)
• D. (3, -6)

Answer 1

The Correct Option is D

To solve the problem, we need to find the third vertex of a triangle given the centroid and two of its vertices.

1. Centroid Formula:
The centroid \( G(x, y) \) of a triangle with vertices \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \) is given by:
\[ G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \]

2. Substituting Known Values:
Given:
Centroid \( G = (4, 1) \)
Vertex A = \( (2, 3) \), Vertex B = \( (7, 6) \)
Let the third vertex be \( C = (x, y) \)

Using the centroid formula:
\[ \frac{2 + 7 + x}{3} = 4 \quad \text{and} \quad \frac{3 + 6 + y}{3} = 1 \]

3. Solving for x:
\[ \frac{9 + x}{3} = 4 \Rightarrow 9 + x = 12 \Rightarrow x = 3 \]

4. Solving for y:
\[ \frac{9 + y}{3} = 1 \Rightarrow 9 + y = 3 \Rightarrow y = -6 \]

5. Coordinates of the Third Vertex:
The third vertex is \( (3, -6) \)

Final Answer:
The third vertex is (3, -6).