# Median

The median is used to refer to a line segment that goes through a vertex to the midpoint on the opposite side. The median bisects the core triangle into 2 smaller triangles of equal area. The median can be represented visually as such:

The length of the median can be determined using the following formulas:

ma $$=$$ sqrt(frac(2b^2 + 2c^2 - a^2)(4)) mb $$=$$ sqrt(frac(2a^2 + 2c^2 - b^2)(4)) mc $$=$$ sqrt(frac(2a^2 + 2b^2 - c^2)(4))

The subscript of $$m$$ determines which side the median is created on. In the first instance, the median was created opposite of vertex $$a$$. The formula can account for vertices $$b$$ and $$c$$ as well by rearranging the variables.

Example

Determine the length of median $$\text{DM}$$ from the given triangle:

We can identify $$\text{EF} = 10$$, $$\text{DF} = 8$$ and $$\text{DE} = 5$$. We can plug the values into the formula to determine the median formed on side EF:

mEF $$= \sqrt{\cfrac{2(5)^2 + 2(8)^2 - (10)^2}{4}}$$

mEF $$= \sqrt{\cfrac{50 + 128 - 100}{4}}$$

mEF $$= \sqrt{\cfrac{78}{4}}$$

mEF $$= \sqrt{19.5} ≅ 4.4$$

Therefore, we can determine that the length of median $$\text{DM}$$ is $$4.4$$.

Determine the length of median $$\text{GM}$$ given the coordinates $$\text{G}(1,2)$$, $$\text{H}(4,1)$$ and $$\text{I}(6,5)$$.