In an equilateral triangle ABC, the centroid is denoted by G. Each side of the triangle is 6 cm long. What is the length (in cm) of the segment AG drawn from vertex A to the centroid G?

Introduction / Context:
This question involves a standard result from triangle geometry, specifically for an equilateral triangle. The centroid G of a triangle is the common intersection point of its medians, and in any triangle it divides each median in the ratio 2 : 1 from vertex to midpoint. For an equilateral triangle, due to its symmetry, the centroid, circumcentre, incentre, and orthocentre all coincide, and the medians have a simple relation to the side length. The problem asks for the distance from a vertex to the centroid in such a triangle.

Given Data / Assumptions:

• Triangle ABC is equilateral.
• Each side of the triangle is 6 cm.
• G is the centroid of triangle ABC.
• The centroid divides each median in the ratio 2 : 1, measured from the vertex.
• We need the length of AG in centimetres.

Concept / Approach:
In any triangle, a median is a line segment joining a vertex to the midpoint of the opposite side. The centroid is the intersection point of the three medians, and it divides each median in the ratio 2 : 1. For an equilateral triangle of side a, the length of any median is equal to the altitude, which can be found using the Pythagoras theorem. Once we know the median length, we use the centroid property to find the distance from the vertex to the centroid, which is two-thirds of the median.

Step-by-Step Solution:
Let the side length a = 6 cm.Consider triangle ABC with median from vertex A to midpoint of BC.The midpoint of BC divides BC into two segments of length 3 cm each.The median from A is also the altitude in an equilateral triangle, so we can use the Pythagoras theorem.Let the median (and altitude) have length m. Then m^2 + 3^2 = 6^2.So m^2 = 36 - 9 = 27.Thus m = √27 = 3√3 cm.The centroid G divides the median in the ratio 2 : 1 from the vertex, so AG = (2 / 3) * m.Therefore, AG = (2 / 3) * 3√3 = 2√3 cm.

Verification / Alternative check:
We can also recall the direct formula for the distance from a vertex to the centroid in any equilateral triangle: AG = (2 / 3) * height, and the height of an equilateral triangle with side a is (√3 / 2) * a. For a = 6, height = (√3 / 2) * 6 = 3√3. Then AG = (2 / 3) * 3√3 = 2√3 cm, which perfectly matches the computed value. This alternative approach confirms the result without re-deriving the median with coordinates.

Why Other Options Are Wrong:
Values such as 2√2 cm, 3√2 cm, and 3√3 cm correspond to incorrect uses of geometric formulas or misapplication of the Pythagoras theorem. For example, 3√3 cm is the full median length, not the distance from the vertex to the centroid. The option 4 cm is a simple but incorrect guess that does not correspond to any standard ratio in an equilateral triangle. Only 2√3 cm correctly represents the length from vertex A to the centroid G.

Common Pitfalls:
Students often confuse the full altitude or median with the vertex-to-centroid distance and choose 3√3 cm instead of 2√3 cm. Another common error is using half the side incorrectly in the Pythagoras calculation. A few learners may also forget that the centroid divides the median in the ratio 2 : 1 and assume that AG is half the median. Remembering the standard ratios and understanding the geometry of special triangles is essential to avoid these mistakes.

Final Answer:
The length of AG in the equilateral triangle of side 6 cm is 2√3 cm.