# Median of a Triangle

## Definition

The median of a triangle is a line segment that joins a vertex to the midpoint of the side that is opposite to that vertex.

## Properties

1. Each triangle has 3 medians, one coming from each vertex; in △ABC, AD, BF, and CE are the 3 medians
2. The 3 medians meet at a common point, called the centroid of the triangle. The centroid is the center of gravity or center of mass of the triangle; the three medians, ma, mb, & mc meet at point O, which is the centroid of the given triangle
3. Each median divides the main triangle into 2 smaller triangles having equal area. Thus, the 3 medians divide the main triangle into 6 smaller triangles having equal area; median AD forms triangles △ABD and △ACD, median BF into △ABF and △CBF, and median CE into △CAE and △CBE

## Formulas

### How to Find the Median of a Triangle

A theorem called Apollonius’s Theorem gives the length of the median of a triangle. According to the theorem: ‘the sum of the squares of any two sides of a triangle equals twice the square on half the third side and twice the square on the median bisecting the third side’.

Accordingly, the standard formulas for finding the medians of a triangle are given below:

Let us solve some examples involving different types of triangles.

### Solved Examples

Find the length of all three medians of a triangle with side 6 cm, 4 cm, and 2 cm.

Solution:

As we know,
m = √2b2 + 2c2 – a2/4, here a = 6 cm, b = 4 cm, and c = 2 cm
Now, putting the values we get,
m = √2(4)2 + 2(2)2 – (6)2/4
= √32 + 8 – 36/4
= √4/4 = 1 cm

In the given triangle, G is the centroid and BC = 8 cm. Solve the value of BL.

Solution:

For △ABC,
AL, BF, and CE are the medians and L is the midpoint of BC
Thus,
BL = 1/2BC
= ½ X 8
= 4 cm

Find the length of the median of an equilateral triangle with sides measuring 9 cm.

Solution:

As we know,
Median (m) = a√3/2, here a = 9 cm
= 9√3/2 [∵ √3= 1.732]
= 7.79 cm