. Centroid of a Triangle

# Centroid

The Centroid is the central point of the triangle which the 3 medians cross over. The centroid lies two-thirds of the distance from each vertex to the midpoint on its opposite side. The medians and centroid can be visually represented as such:

Using the points given by the centroid, the length of the median can be determined using the following formula(s), referred to as the Segment Addition Postulate:

$$\text{XM} = \text{XC} + \text{MC}$$
$$\text{XM} = \cfrac{2}{3}\text{XM} + \cfrac{1}{3}\text{XM}$$

Example

Determine the respective lengths of $$\text{PC}$$ and $$\text{MC}$$ if the length of $$\text{PM}$$ is $$30$$.

We can use Segment Addition Postulate to determine the missing side lengths:

$$\text{PC} = \cfrac{2}{3}\text{PM}$$
$$\text{PC} = \cfrac{2}{3}(30) = 20$$
$$\text{PC} = 20$$

$$\text{MC} = \text{PM} - \text{PC}$$
$$\text{MC} = 30 - 20 = 10$$
$$\text{MC} = 10$$

Therefore, we can determine that $$\text{PC}$$ and $$\text{MC}$$ have respective side lengths of $$20$$ and $$10$$.

Determine the respective lengths of $$\text{TM}$$ and $$\text{MC}$$ if the length of $$\text{TC}$$ is $$8$$