The centroid of a triangle is a point of trisection of each of its medians. True or False?
The correct answer and explanation is:
Correct Answer: True
The centroid of a triangle is indeed a point of trisection of each of its medians.
Explanation (Approx. 300 words):
In geometry, the centroid is the point where all three medians of a triangle intersect. A median is a line segment that connects a vertex of the triangle to the midpoint of the opposite side. Every triangle has exactly three medians, and they always intersect at a single point — the centroid.
Now, here’s the key detail: the centroid divides each median in the ratio 2:1, where the longer segment is between the vertex and the centroid, and the shorter is between the centroid and the midpoint of the side. This means that the centroid is located two-thirds of the way along each median, starting from the vertex.
This division implies that the centroid trisects each median — it divides it into two parts, one of which is twice as long as the other. Therefore, the centroid is not located halfway along the median (as a midpoint would be), but at a point of trisection: one part is 2 units long, the other is 1 unit long, totaling 3 units.
Example:
Let’s say you have a triangle with vertices A, B, and C. Draw the median from vertex A to the midpoint M of side BC. The centroid G lies on this median such that:
- AG (from vertex A to centroid G) is 2/3 of the median’s length,
- GM (from centroid G to midpoint M) is 1/3 of the median’s length.
This property is extremely useful in coordinate geometry, physics (center of mass), and engineering.
Conclusion:
Since the centroid divides each median in a 2:1 ratio and lies at a point of trisection, the statement is True.