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:
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 \(EF\):
\(m_{\text{EF}} = \sqrt{\cfrac{2(5)^2 + 2(8)^2 - (10)^2}{4}}\)
\(m_{\text{EF}} = \sqrt{\cfrac{50 + 128 - 100}{4}}\)
\(m_{\text{EF}} = \sqrt{\cfrac{78}{4}}\)
\(m_{\text{EF}} = \sqrt{19.5} \approx 4.4\)
Therefore, we can determine that the length of median \(\text{DM}\) is approximately \(\boldsymbol{4.4}\).
First, we can draw a diagram of the triangle to help visualize the problem:
Next, we can determine the location of \(M\), which is the midpoint of \(HI\), by using the midpoint formula:
\(\text{midpoint} = \left(\cfrac{4 + 6}{2}, \cfrac{1 + 5}{2}\right)\)
\(\text{midpoint} = \left(\cfrac{10}{2}, \cfrac{6}{2}\right)\)
\(\text{midpoint} = (5, 3)\)
Using \(M\), we can determine the length of \(GM\) by using distance formula:
\(d_{\text{GM}} = \sqrt{(5 - 1)^2 + (3 - 2)^2}\)
\(d_{\text{GM}} = \sqrt{(4)^2 + (1)^2}\)
\(d_{\text{GM}} = \sqrt{16 + 1}\)
\(d_{\text{GM}} = \sqrt{17} \approx 4.1\)
Therefore, we can determine that the length of median \(\text{GM}\) is approximately \(\boldsymbol{4.1}\).