What is the Median of a Triangle?

A median of a triangle runs from a vertex to the midpoint of the opposite side. Every triangle has three, and each one divides the triangle into two smaller triangles of equal area.

• Centroid Division: The centroid divides each median into two segments, with the segment connecting the centroid to the vertex being twice as long as the segment connecting the centroid to the midpoint of the side.
• Concurrent Medians: All three medians of a triangle are concurrent, meaning they meet at one point, the centroid. This point also serves as the triangle’s center of gravity.
• Area Relation: The medians of a triangle divide it into six smaller triangles of equal area.

Worked examples: the median of a triangle

Practice questions on triangle medians

1. If the median of a triangle is (18 text{ cm} ) long, how far is the centroid from the triangle’s vertex?
2. In a triangle, the distance from a vertex to the centroid along a median is (9 text{ cm} ). What is the total length of this median?

1. ( frac{2}{3} times 18 text{ cm} = 12 text{ cm} )
2. Given the (2:1) ratio, the total median length (= 9 text{ cm} times 3 = 27 text{ cm} )

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