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:
\(\text{XM} = \text{XC} + \text{MC}\)
\(XM = \cfrac{2}{3}\text{XM} + \cfrac{1}{3}\text{XM}\)
ExampleDetermine the respective lengths of \(\text{PC}\) and \(\text{MC}\) if the length of \(\text{PM}\) is \(30\).
First, we can use Segment Addition Postulate to determine the side length of \(\text{PC}\):
\(\text{PC} = \cfrac{2}{3}\text{PM}\)
\(\text{PC} = \cfrac{2}{3}(30)\)
\(\text{PC} = 20\)
Next, we can determine the side length of \(\text{MC}\) by calculating the difference between sides \(\text{PM}\) and \(\text{PC}\):
\(\text{MC} = \text{PM} - \text{PC}\)
\(\text{MC} = 30 - 20\)
\(\text{MC} = 10\)
Therefore, we can determine that \(\textcolor{red}{\text{PC}}\) and \(\textcolor{blue}{\text{MC}}\) have respective side lengths of \(\boldsymbol{\textcolor{red}{20}}\) and \(\boldsymbol{\textcolor{blue}{10}}\).
First, we can can rearrange the Segment Addition Postulate formula to determine the missing side lengths:
\(\text{TC} = \cfrac{2}{3}\text{TM}\)
\(8 = \cfrac{2}{3}\text{TM}\)
\(\cfrac{3}{2}(8) = \cfrac{3}{2}\left(\cfrac{2}{3}\text{TM}\right)\)
\(\text{TM} = 12\)
Next, we can determine the side length of \(\text{MC}\) by calculating the difference between sides \(\text{TM}\) and \(\text{TC}\):
\(\text{MC} = \text{TM} - \text{TC}\)
\(\text{MC} = 12 - 8\)
\(\text{MC} = 4\)
Therefore, we can determine that \(\textcolor{red}{\text{TM}}\) and \(\textcolor{blue}{\text{MC}}\) have respective side lengths of \(\boldsymbol{\textcolor{red}{12}}\) and \(\boldsymbol{\textcolor{blue}{4}}\).