Median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. Medians can be found inside the triangle and a triangle can have three medians. Irrespective of the shape the three medians will intersect at a single point known as the centroid of the triangle.

Given below are the properties of medians of triangles.
  • Each median divides the triangle into two smaller triangles resulting in same area.
  • If the triangles are of different shapes then the three medians will divide the 6 smaller triangles all having same area.
  • Medians are concurrent
  • Medians of a triangle intersect each other in the ratio 2 : 1.
  • Centroid will be the center of gravity of the triangle.
  • Length of the medians are defined by the following equations
  1. $m_{i}$ = $\frac{1}{2}$ $\sqrt{j^{2} +k^{2} + 2jk \cos i}$
  2. $m_{j}$ = $\frac{1}{2}$ $\sqrt{i^{2} +k^{2} + 2jk \cos j}$
  3. $m_{k}$ = $\frac{1}{2}$ $\sqrt{j^{2} +i^{2} + 2jk \cos k}$
  • In terms of median area of a triangle can be expressed as
A = $\frac{4}{3}$ $\sqrt{s_{m}(s_{m} - m_{1}) (s_{m} - m_{2}) (s_{m} - m_{3})}$
where $S_{m}$ = $\frac{1}{2}$ $(m_{1} + m_{2} + m_{3})$
The formula to find the median of a triangle is given by:

$m_{a}$ = $\sqrt{\frac{2b^{2}+2c^{2} - a^{2}}{4}} $
where a, b and c will be the sides of the triangle having their respective medians as $m_{a}$, $m_{b}$ and $m_{c}$ from their midpoints.
Lines containing the same point are said to be concurrent and point of concurrency is the place where three or more lines intersect at the same point. Median of triangles are said to be concurrent as the concurrence will be in the interior of the triangles.
Centroid of Triangle
A point where the medians concurrent is the centroid of a triangle and the centroid dividing the median will be in the ratio 2 :1.

Solved Examples

Question 1: In the triangle if the sides are a = 8, b = 9 and c = 5. Find the length of the median drawn to side 'a'.
Solution:
 
Given a = 8, b = 9 and c = 5
The formula to find median is given as

$m_{a}$ = $\sqrt{\frac{2b^{2}+2c^{2} - a^{2}}{4}} $

Plug in the given values to solve the problem.

$m_{a}$ = $\sqrt{\frac{2(9)^{2}+2(5)^{2} - 8^{2}}{4}} $

$m_{a}$ = $\sqrt{\frac{162 + 50 - 64}{4}} $

$m_{a}$ = $\sqrt{37} $

$m_{a}$ = 6.08
 

Question 2: In the triangle if the sides are a = 4, b = 5 and c = 7. Find the length of the median drawn to side 'c'.
Solution:
 
Given a = 4, b = 5 and c = 7
The formula to find median is given as

$m_{c}$ = $\sqrt{\frac{2b^{2}+2a^{2} - c^{2}}{4}} $

Plug in the given values to solve the problem.

$m_{c}$ = $\sqrt{\frac{2(5)^{2}+2(4)^{2} - 7^{2}}{4}} $

$m_{c}$ = $\sqrt{\frac{50 + 32 - 49}{4}} $

$m_{c}$ = $\sqrt{8.25} $

$m_{c}$ = 2.87