Median of a triangle
The median of a triangle is a line segment that joins one of its vertices with the center of the opposite side.
Every triangle has three medians (ma, mb and mc), one from each vertex. The lengths of the medians can be obtained from Apollonius’ theorem as:
Where a, b, and c are the sides of the triangle with respective medians ma, mb and mc from their midpoints.
In any median of a triangle, the distance between the center of gravity (or centroid) G and the center of its corresponding side is one third (1/3) of the length of that median, i.e., the centroid is two thirds (2/3) of the distance from each vertex to the midpoint of the opposite side.
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Note. Courtesy of the author: José María Pareja Marcano. Chemist. Seville, Spain.
In a triangle, it is true that the sum of the squares of two of its sides is equal to the sum of half the square of the third side and twice the square of the median corresponding to this third side.
Now, in a triangle Δ ABC, we have:
Where a, b, and c, are the legs and mb is the median corresponding to side b.
Find the length of the median of a triangle Δ ABC if length of sides are a=2 cm, b=4 cm and c=3 cm.
Using the equation of Apollonius’ theorem, we can find the length of all three medians:
Thus, the medians are ma=3.39 cm, mb=1.58 cm and mc=2.78 cm.
The medians of a triangle are the line segments created by joining one vertex to the midpoint of the opposite side. Since every triangle has three sides and three angles, it has three medians (ma, mb and mc).
Centroid theorem: the distance between the centroid and its corresponding vertex is twice the distance between the barycenter and the midpoint of the opposite side. That is, the distance from the centroid to each vertex is 2/3 the length of each median. This is true for every triangle.
The centroid is always inside the triangle.
AUTHOR: Bernat Requena Serra