Question

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

• A. (5, 2)
  (B)
  (C)
  (D)
  \textbf{Solution:}
  \vspace{0.5cm} The centroid of a triangle is found by taking the average of the coordinates of its three vertices. This point represents the balance point of the triangle. \vspace{0.5cm} \textbf{Topic - Triangle and Centroid} \vspace{0.5cm} \textbf{ In } \( \triangle ABC \), \( AD \) is the bisector of \( \angle ABC \), \( AB = 4 \, \text{cm} \), \( AC = 6 \, \text{cm} \), and \( BD = 2 \, \text{cm} \). Find the value of \( DC \).
• B. (1, 3)
• C. (6, 0)
• D. (2, -3)

Hint:

The centroid of a triangle is found by taking the average of the coordinates of its three vertices. This point represents the balance point of the triangle.

Answer 1

The Correct Option is C

The formula for the centroid is:
\[ \text{Centroid} = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \] where \(A(x_1, y_1) = (2, 3)\), \(B(x_2, y_2) = (1, -3)\), and Centroid \(C(x_3, y_3) = (3, 0)\) is given. \[ \frac{2 + 1 + x_3}{3} = 3 \quad \text{and} \quad \frac{3 - 3 + y_3}{3} = 0 \] From the first equation: \[ \frac{3 + x_3}{3} = 3 \quad \Rightarrow \quad 3 + x_3 = 9 \quad \Rightarrow \quad x_3 = 6 \] From the second equation: \[ \frac{3 + y_3}{3} = 0 \quad \Rightarrow \quad 3 + y_3 = 0 \quad \Rightarrow \quad y_3 = -3 \] Therefore, the coordinates of the third vertex C are \((6, 0)\). Correct Answer: (C) (6, 0)