In equilateral triangle ABC, side AB = 12 cm. O is the centroid of the triangle (intersection of medians). What is the length (in cm) of segment AO?

Introduction / Context:
This problem involves an equilateral triangle and its centroid. In an equilateral triangle, several important line segments coincide: medians, angle bisectors, perpendicular bisectors and altitudes. The centroid is a key point used in geometry and physics (centre of mass), and its distance from a vertex can be expressed in terms of the side length.

Given Data / Assumptions:

• Triangle ABC is equilateral.
• Side AB = 12 cm (so all sides are 12 cm).
• O is the centroid, the point where the three medians intersect.
• We must find the length AO.
• In any triangle, the centroid divides each median in the ratio 2 : 1 from the vertex.

Concept / Approach:
First, we find the length of a median in an equilateral triangle. The median is also the height. For an equilateral triangle of side s, the height h is: h = (√3 / 2) * s The centroid lies two thirds of the way from the vertex to the midpoint of the opposite side along the median. Therefore: AO = (2 / 3) * h We substitute s = 12 cm into these formulas to find AO.

Step-by-Step Solution:
Step 1: Let the side length s of the equilateral triangle be 12 cm. Step 2: Compute the height h of the equilateral triangle: h = (√3 / 2) * s. Step 3: Substitute s = 12 to get h = (√3 / 2) * 12 = 6√3 cm. Step 4: The centroid divides the median in the ratio 2 : 1 from the vertex, so AO = (2 / 3) * h. Step 5: Substitute h = 6√3: AO = (2 / 3) * 6√3 = 4√3 cm. Step 6: Therefore, AO = 4√3 cm.

Verification / Alternative check:
We can check the lengths of the segments on the median. If AO = 4√3 cm, then the remaining segment from O to the midpoint of BC is (1 / 3) * h = 2√3 cm. The total median length is AO + OM = 4√3 + 2√3 = 6√3, which matches the computed height. Also, using coordinate geometry with the triangle placed conveniently and computing distances confirms that AO equals 4√3 cm for side length 12 cm.

Why Other Options Are Wrong:
2√3 cm is the distance from the centroid to the midpoint of a side (one third of the median), not from the vertex. 3√3 cm and 9√3 cm do not match the 2 : 1 division of the median and give an incorrect total height. 6√3 cm is the full median or height, not the vertex to centroid distance. Only 4√3 cm is exactly two thirds of the median length and fits the centroid division property.

Common Pitfalls:
Some students mistakenly think the centroid divides the median in half, rather than in the ratio 2 : 1. Others confuse the incenter or circumcenter with the centroid, even though in an equilateral triangle these points coincide but are approached through different formulas. Remember that in any triangle the centroid always splits each median into a longer segment (2 parts) from vertex to centroid and a shorter segment (1 part) from centroid to side midpoint.

Final Answer:
The length of AO is 4√3 cm.