The Centroid and Its Role in Triangles

The centroid is where a triangle three medians meet, and it is the triangle balance point. It sits two thirds of the way along each median from the vertex, which is the ratio most problems turn on.

• Division of Medians: The centroid divides each median in a \(2:1\) ratio, with the segment from the vertex to the centroid being twice as long as the segment from the centroid to the midpoint of the side.
• Center of Gravity: The centroid acts as the center of gravity or the balance point of a triangle. If a triangle were to be made out of a uniform material, it would balance perfectly on its centroid.
• Area Distribution: When the medians intersect at the centroid, they divide the triangle into six smaller triangles. Interestingly, all these triangles have equal areas.

Worked examples: finding the centroid

Practice questions on the centroid of a triangle

1. In a triangle, if the median’s length is \(24\text{ cm} \), how far is the centroid from the midpoint of the side the median is drawn to?
2. If the distance from a vertex to the centroid along a median in a triangle is \(15\text{ cm} \), what is the total length of this median?

1. \( \frac{1}{3} \times 24 \text{ cm} = 8 \text{ cm} \)
2. Given the \(2:1\) ratio, the total median length \(= 15 \text{ cm} \times \frac{3}{2} = 22.5 \text{ cm} \)

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