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 \(8\)
Show Answer
We can rearrange the Segment Addition Postulate formula to determine the missing side lengths:
\(\text{TC} = \cfrac{2}{3}\text{TM}\)
\((\cfrac{3}{2}(8) = \cfrac{3}{2}(\cfrac{2}{3}\text{TM})\)
\(\text{TM} = 12\)
\(\text{MC} = \text{TM} - \text{TC}\)
\(\text{MC} = 12 - 8 = 4\)
\(\text{MC} = 4\)
Therefore, we can determine that \(\text{TM}\) and \(\text{MC}\) have respective side lengths of \(12\) and \(4\).