Median

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.

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.

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.

• Between the centroid and the interior angle are two parts of the median.

• Between the centroid and the opposite side is the rest of the median.

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 \(2:1\).

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 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.

This brings us to the end of the article here are the key takeaways to refresh what we've learned so far.