The Centroid and Its Role in Triangles

• 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.

Examples

Practice Questions:

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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