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:
- 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?
- 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?
- \( \frac{1}{3} \times 24 \text{ cm} = 8 \text{ cm} \)
- Given the \(2:1\) ratio, the total median length \(= 15 \text{ cm} \times \frac{3}{2} = 22.5 \text{ cm} \)
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Original price was: $109.99.$54.99Current price is: $54.99.
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