Median is a line drawn from a vertex to the midpoint of the opposite side. Median divides a triangle in two parts of equal areas. These two parts may not be similar but they are equal in areas. In the triangle shown below \(AD, BE\) and \(CF\) are medians. These medians intersect at a common point \(G\), known as centroid of the triangle. 

Also medians intersect each other in the ratio 2 : 1.

\[\frac{{AG}}{{GD}} = \frac {{BG}}{{GE}} = \frac {{CG}}{{GF}} = \frac {2}{1}\]

Length of a Median

In the triangle \(ABC\) (shown above), \(D\) is the midpoint and \(AD\) is the median. 

By Apollonius theorem length of \(AD\) is given by 

\[\bbox[5px, border: 2px solid #0071dc]{\color{blue}{A{B^2} + A{C^2} = {\rm{ }}2{\rm{ }}\left( {A{D^2} + D{C^2}} \right)}}\]This formula can be applied to calculate median when all three sides of a triangle are given.

Example  01: In a triangle ABC, AB = 13, AC = 15 and BC = 14. Find the length of the median AD.

Medians of a right angled triangle:

Let \(ABC\) is a right angled triangle with right angle at \(B\), then its shortest median is \(BD\) and is equal to half of the hypotenuse \(AC\). Actually midpoint of hypotenuse of any right angled triangle is equidistance from the three vertices.

If the medians are \({m_1},\,{m_2}\) and \({m_3}\), where \({m_3}\) is the shortest median then \(AC = 2{m_3}\) and

\[{m_2}^2 = A{B^2} + {\left( {\frac{{BC}}{2}} \right)^2}\]

\[m_1^2 = B{C^2} + {\left( {\frac{{AB}}{2}} \right)^2}\]

Adding both the equations, we have

\[m_{_1}^2 + m_2^2 = A{B^2} + \frac{{B{C^2}}}{4} + B{C^2} + \frac{{A{C^2}}}{4} = \frac{5}{4}\left( {A{B^2} + B{C^2}} \right)\]

\[ \Rightarrow m_1^2 + m_2^2 = \frac{5}{4}\left( {2m_3^2} \right)\]

\[\color{blue}{\Rightarrow  m_1^2 + m_2^2 = 5m_3^2}\]

Example  02: In a right angled triangle, one of the median is 10 and the hypotenuse is 16. Find the length of the length of the longest median.