Suppose you have to share the last cake piece with your brother. And neither one of you wants to get a smaller piece. To avoid the fight between you and your brother over a cake your mother cuts the triangular cake piece from its median so both of you get the same size of cake. But what is this median? How did your mother decide where to cut the cake?
We define the median of a triangle as the line segment joining the vertex to the midpoint of its opposite side. In this article, we will go through the definition of a median, its different properties, the mathematical formula & finally work through a few examples.
At the end of this article, you will be able to:
Define a median and relate it to the area of a triangle.
Identify and draw medians in a triangle.
Calculate the length of the median from the sides and the coordinates of a triangle.
Median meaning
So, what exactly does the median mean? Imagine you have a slice of pizza that you need to share between yourself and your friend. For simplicity, let's call that pizza \(\bigtriangleup ABC\). Now keep in mind that you need to divide the pizza equally among your friends. Here's where the median can help.
Median of pizza slice, pexels.com
Choose a side of the pizza say side \(a\) (i.e. side \(BC\)), and cut the pizza across the line segment joining the midpoint of the line and the opposite interior angle, as shown in the figure below. Hurrah! You and your friend can now enjoy equal amounts of pizza. The imaginary line that cut the pizza into two equal parts is the median. Since all triangles have \(3\) sides and \(3\) interior angles. It will always have \(3\) medians.
A median is a constructed line that connects the midpoint of one side to the opposite interior angle.
It is interesting to note that a triangle's perimeter is always larger than the sum of its three medians.
What is a centroid?
Now that we know what a median is, let's explore what is a centroid. The point where the 3 medians cross is called the centroid. The centroid is a point of concurrency. A point of concurrency is a point where two or more lines intersect. Such as the point of intersection of medians, perpendicular bisectors, and altitudes. The centroid will always lie inside the triangle unlike other points of concurrency.
The point where the three medians intersect is called the centroid.
Fig. 1. Three medians with centroid as intersecting point.
The centroid has a few interesting properties. It will always divide the median into a \(2:1\) ratio. The centroid is always located two-thirds along the median from the interior angle.
Let us visualize how the median is divided into a ratio of \(2:1\). Take a \(\bigtriangleup ABC\) and draw the \(3\) medians from each of the vertexes. Now let \(O\) be the centroid of the triangle. If \(AM\) is the median of the triangle from vertex \(A\) then \(2OM = OA\).
Fig. 2. Centroid divides the median into parts of \(2:1\).
In a right-angled triangle, the median drawn from the right angle's vertex is half the length of the triangle's hypotenuse. The median from the right angle of the triangle cuts the hypotenuse into two equal parts and each part of the hypotenuse is equal to the length of the median.
Fig. 3. Median equal to half of the hypotenuse.
In the above figure, The median \(AD\) cuts the hypotenuse into two equal parts \(CD\) and \(BD\), in such a way that \(AD=CD; AD=BD\).
Properties of median
The properties of a median can be described as follows:
Any triangle contains 3 medians with an intersecting point called the centroid.
The connecting side of the median is divided into two equal parts.
Two Triangles of equal size and area are formed by constructing a median from any of the vertices of a triangle.
In fact, any triangle is divided into 6 smaller Triangles with equal area by 3 medians of the triangle.
Median and Altitude of a triangle
It can be a bit confusing to distinguish between the median and Altitude of a triangle and easy to consider both of them the same. But the median and Altitude of a triangle are two different elements of a triangle. The median of a triangle is a line segment from one vertex to the mid-point of its opposite side. Whereas, the altitude of a triangle is a perpendicular line segment from a vertex to its opposite side.
Fig. 4. Median and Altitude of triangles.
In the above figure, \(AD, BE,\) and \(CF\) are the medians of the triangle \(\bigtriangleup ABC\), and \(XM, YN,\) and \(ZO\) are the altitudes of the triangle \(\bigtriangleup XYZ\).
Difference between median and altitude
Let's see the difference between the median and altitude of a triangle.
| Median | Altitude |
- A median is a line segment from a midpoint of one side to its opposite vertex.
| - An altitude is a perpendicular line segment formed from one side to its opposite vertex point.
|
- A triangle has 3 medians with an intersecting point called the centroid.
| - A triangle has 3 altitudes with an intersecting point called the orthocenter.
|
- All the 3 medians are inside the triangle of any shape.
| - Altitude may or may not be located inside the triangle depending upon the shape.
|
- A median divides a triangle into two smaller triangles with equal area.
| - An altitude divides a triangle into two smaller triangles, but their area might not be same.
|
Median of the triangle formula
A basic formula can be used to calculate the median of a triangle. Let's look at the median of the triangle formula to calculate the length of each median.
Fig. 5. Three median of a triangle.
The formula for the first median is as follows:
\[m_a=\sqrt{\frac{2b^{2}+2c^{2}-a^{2}}{4}}\]
where \(a, b,\) and \(c\) are the side lengths, and \(m_a\) is the median from interior angle \(A\) to side \('a'\).
The formula for calculating the second median of a triangle is as follows:
\[m_b=\sqrt{\frac{2c^{2}+2a^{2}-b^{2}}{4}}\]
where the triangle's median is \(m_b\), the sides are \(a, b,\) and \(c\), and the median is formed on side \('b'\).
Similarly, The formula for the third median of a triangle is as follows:
\[m_c=\sqrt{\frac{2a^{2}+2b^{2}-c^{2}}{4}}\]
where the triangle's median is \(m_c\), the triangle's sides are \(a, b,\) and \(c\), and the median is formed on side \('c'\).
But then how would we calculate the length using only the coordinates of the triangle? We first estimate the midpoints of the side with the median using the formula below.
Fig. 6. Triangle with midpoint and median.
\[M(x_m, y_m)=\frac{(x_2+x_3)}{2}, \frac{(y_2+y_3)}{2}\]
where \(M(x_m,y_m)\) is one of the endpoints of the median. Using these coordinates and the remaining point, we can calculate the length of the median. The coordinates need to be substituted into the following formula. This is the distance formula and it gives the distance between any two coordinates on a 2-D plane.
\[D=\sqrt{(x_m-x_1)^2+(y_m-y_1)^2}\]
where \(D\) is the distance between the two points and \((x_1,y_1) , (x_m,y_m)\) are the coordinates of the endpoints of the median.
Similarly, to calculate the distance between \((x_2,y_2)\) and the midpoint of the opposite side \(AC\) we use,
\[D=\sqrt{(x_m-x_2)^2+(y_m-y_2)^2}\]
where \(M(x_m, y_m)=\frac{(x_1+x_3)}{2}, \frac{(y_1+y_3)}{2}\).
And the distance between \((x_3,y_3)\) and the midpoint of the opposite side \(AB\) is calculated using,
\[D=\sqrt{(x_m-x_3)^2+(y_m-y_3)^2}\]
where \(M(x_m, y_m)=\frac{(x_1+x_2)}{2}, \frac{(y_1+y_2)}{2}\).
Median examples
Let's take a look at some median examples and understand it.
Find the length of the median of the given triangle \(ABC\) whose sides are given as follows, \(AB = 10\, units\), \(BC = 6\, units\), and \(AC = 8\, units\), respectively, in which AM is the median formed on side \(BC\).
Fig. 7. Triangle with side lengths.
Solution:
\[AM=\sqrt{\frac{2AB^2+2AC^2-BC^2}{4}}\]
where \(AB=10, BC=6, AC=8\)
Substituting the values in the formula, we will get, Median \(AM\),
\[AM=\sqrt{\frac{(2\times 10^2)+(2\times 8^2)-6^2}{4}} = 8.54\]
Hence, the length of median \(AM\) is \(8.54 \; units\).
Find the length of the median \(AM\) if the coordinates of the triangle \(ABC\) are given as, \(A (2,5), B (6,3), C (-3,0)\).
Fig. 8. Triangle with coordinates.
Solution:
Step 1: Calculate the coordinates of the midpoint for \(BC\)
\begin{align}M(x,y)&=\frac{(x_1+x_2)}{2}, \frac{(y_1+y_2)}{2} \\&=\frac{(6+(-3))}{2}, \frac{3+0}{2} \\&=(1.5, 1.5)\end{align}
Step 2: Now that we have the coordinates for the endpoints of the median we can estimate the length using the distance formula.
\begin{align}d&=\sqrt{(1.5-2)^2+(1.5-5)^2} \\&=\sqrt{(-0.5)^2+(-3.5)^2} \\&=\sqrt{12.5} \\&=3.53\end{align}
This gives us the length of the \(3.53\) units.
This brings us to the end of the article here are the key takeaways to refresh what we've learned so far.
Median - Key takeaways
- The median is a line segment joining the vertex and the midpoint of the opposite side.
- It divides the opposite side into two equal parts by bisecting it.
- The median divides the triangle into two triangles of equal areas. The \(3\) medians will divide the triangle into \(6\) equal triangles.
- Each triangle has three medians, the point of intersection of the medians is called the centroid.
- The length of the median can be calculated using the formula: \[m_a=\sqrt{\frac{2b^{2}+2c^{2}-a^{2}}{4}}\].
- The length can also be calculated using the coordinates of the triangles using the distance formula and the midpoints of the opposite side. \(M(x_m, y_m)=\frac{(x_2+x_3)}{2}, \frac{(y_2+y_3)}{2}\), where M is the midpoint \(D=\sqrt{(x_m-x_1)^2+(y_m-y_1)^2}\), where \(D\) is the distance.
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