Question

If coordinates of the circumcentre and the orthocentre of a triangle are respectively \( (5,5) \) and \( (2,2) \), then the coordinates of the centroid are:

• A. \( (1,1) \)
• B. \( (3,1) \)
• C. \( (3,3) \)
• D. \( (2,2) \)
• E. \( (4,4) \)

Hint:

Use the relation \( G = \frac{2O + H}{3} \) directly to avoid confusion in section formula.

Answer 1

Answer: (E)

Concept:
• Circumcentre \(O\), centroid \(G\), and orthocentre \(H\) lie on the Euler line.
• The centroid divides the line joining \(O\) and \(H\) internally in the ratio \(1:2\), i.e., \[ OG : GH = 1 : 2 \]
• Hence, using section formula: \[ G = \frac{2O + H}{3} \]

Step 1: Identify given points
\[ O = (5,5), \quad H = (2,2) \]

Step 2: Apply centroid formula
\[ G = \left( \frac{2x_O + x_H}{3}, \frac{2y_O + y_H}{3} \right) \]

Step 3: Substitute values
\[ G = \left( \frac{2 \cdot 5 + 2}{3}, \frac{2 \cdot 5 + 2}{3} \right) \]

Step 4: Simplify
\[ G = \left( \frac{12}{3}, \frac{12}{3} \right) \] \[ G = (4,4) \]

Step 5: Final Answer
\[ \boxed{(4,4)} \]