In an equilateral triangle ABC, G is the centroid. If each side of the triangle is 9 cm (so AB = BC = CA = 9 cm), then what is the length of segment AG (in cm)?

Introduction / Context:
This question deals with special properties of an equilateral triangle and its centroid. The centroid is the point where the three medians of a triangle intersect. In any triangle, the centroid divides each median in a 2 : 1 ratio, counting from the vertex to the midpoint of the base. In an equilateral triangle, medians, angle bisectors, perpendicular bisectors and altitudes all coincide, giving the triangle a high degree of symmetry. Using these facts, we can find the length from a vertex to the centroid when we know the side length.

Given Data / Assumptions:
• Triangle ABC is equilateral: AB = BC = CA = 9 cm.
• G is the centroid of triangle ABC, where the three medians intersect.
• In an equilateral triangle, each median is also an altitude and angle bisector.
• We are required to find the distance AG in centimetres.

Concept / Approach:
First, we compute the length of a median (or altitude) of the equilateral triangle. For an equilateral triangle with side a, the length of the altitude is given by (√3 / 2) * a. This altitude is also a median from a vertex to the midpoint of the opposite side. The centroid divides this median in the ratio 2 : 1, with the longer segment adjoining the vertex. Hence, the length from the vertex to the centroid is 2 / 3 of the entire median length. By combining these facts, we can express AG in terms of the side length a and then substitute a = 9 cm.

Step-by-Step Solution:
Step 1: Let the side length of the equilateral triangle be a. Here, a = 9 cm. Step 2: Recall the formula for the altitude (which is also a median) of an equilateral triangle: altitude = (√3 / 2) * a. Step 3: Substitute a = 9 cm into the formula to get the median length: median = (√3 / 2) * 9 = (9√3) / 2 cm. Step 4: In any triangle, the centroid divides each median in the ratio 2 : 1, from the vertex to the midpoint of the opposite side. Step 5: Therefore, AG, which is the segment from vertex A to centroid G, is 2 / 3 of the full median: AG = (2 / 3) * median. Step 6: Substitute the median length: AG = (2 / 3) * (9√3 / 2) = (2 * 9√3) / (3 * 2) = 3√3 cm.

Verification / Alternative check:
We can also check the numeric value of 3√3 cm. Since √3 is approximately 1.732, 3√3 ≈ 3 * 1.732 ≈ 5.196 cm. This is slightly more than half of the side length, which makes sense because the centroid is located inside the triangle, closer to the base than to the midpoint of the altitude. Furthermore, if the entire median is (9√3) / 2 ≈ 7.794 cm, then 2 / 3 of this is indeed about 5.196 cm, consistent with our computation of AG.

Why Other Options Are Wrong:
If AG were 3 cm, that would be significantly smaller than 2 / 3 of the median and would contradict the 2 : 1 median division rule. The option 3√3 / 2 cm corresponds to one third of the median length, not two thirds, so it would place the point closer to the base than to the vertex, which is not the centroid. A length of 6 cm would exceed 2 / 3 of the median and even approach the full median, which is not consistent with the centroid location. Thus these options do not match the known geometric relationships.

Common Pitfalls:
A common mistake is to confuse the centroid with the circumcentre or incentre and use incorrect distances or ratios. Another error is to use the side length directly instead of first computing the median or altitude. Some learners also invert the ratio and take AG as 1 / 3 of the median rather than 2 / 3. To avoid such errors, remember that the centroid is the balancing point of the triangle and lies two thirds of the way from each vertex along each median, which is a fundamental and frequently tested property.

Final Answer:
The length of AG is 3√3 cm.