Median of a triangle is a line segment joining a vertex of a triangle to the midpoint of the opposite side. A triangle has three sides, so every triangle has exactly three medians, each running from one vertex to the side exactly opposite.
In the triangle ABC, D,E and F are the mid-points of sides BC, CA and AB respectively. The line segments AD, BE and CF are the medians of the triangle.
There are some basic properties of medians which make them very important. They are as follows:
(1) Each median divides the triangle into two parts, and the area of these two parts are exactly equal.
In this triangle ABC, the median CM divides the triangle into two parts: ACM and BCM. These two triangles ACM and BCM have area equal to each other.
(2) The three medians of a triangle are concurrent. i.e. they meet at a single point known as the Centroid.
The medians AL, BM and CN meet at P, the centroid of the triangle. This can be proved by the corollary of Ceva's Theorem.
(3) The centroid is the center of gravity of the triangle.
In the above figure, G is the centroid as well as the center of gravity.
(4) The medians divide each other in the ratio 2:1 at the centroid.
In the above figure, G is the centroid, and it divides the median in the ratio 2:1.
One can determine the length of the median of a triangle by a formula known as Apollonius Formula. The formula is given as:
where a, b and c are the lengths of the three sides of the triangle and m is the length of the median connecting side a to vertex A.