Question

Let \( G \) be the centroid of a triangle ABC and \( O \) be any other point in that plane, then \[ OA + OB + OC + OG = \]

• A. \( 4OG \)
• B. \( \vec{0} \)
• C. \( 3OG \)
• D. \( 2OG \)

Hint:

The centroid divides each median into a 2:1 ratio. The sum of the position vectors of any point and the centroid equals four times the vector from the centroid to the point.

Answer 1

The Correct Option is A

Step 1: Understand the centroid property.
The centroid \( G \) of a triangle divides each median in the ratio 2:1. The position vector of the centroid is the average of the position vectors of the vertices. The sum of the position vectors of any point and the centroid is 4 times the vector from the centroid to the point.

Step 2: Conclusion.
Thus, the sum \( OA + OB + OC + OG = 4OG \), corresponding to option (A).