Median of a Triangle

The study of Euclidean flat-space geometry, starts with the concept of a point, line, and segment. Then, we learn about the intersection of lines and segments and get introduced to angles.

This basic mathematics course is followed by an introduction to closed two-dimensional geometrical figures formed from joined segments like triangles, quadrilaterals, and polygons. There are several interesting properties associated with each of them.

A triangle is made up of three segments joined together to form a closed two-dimensional object with three angles (hence the name Tri-Angle) and three vertices. Every triangle has a side facing every one of its vertices. A median is a segment connecting a vertex of a triangle with the midpoint of the opposite side.

If you draw all the medians, they all intersect at the same point, which is called the centroid of that triangle. This property is known as 'concurrency'. No matter what triangle it is (right-angled, isosceles, or scalene), all medians meet at the centroid.

The procedure for finding the median is simple and all it takes is accurate measurement and drawing. There is no better way to find it, other than actually drawing it. The proof of its existence is through actual construction, like most other geometrical concepts.

To draw the median from any vertex, you need to first draw the triangle with the right dimensions. Then, you need to measure out the midpoints of every side and mark them with a point. Once you have done that, all that remains to be done is to use a scale and pencil to connect the vertices with the midpoints of opposite sides, by drawing segments.

The point where the three medians meet, should be labeled as the centroid of the triangle. There are three interesting features associated with medians and their centroid.

One is the fact that it's always located within the triangle and the second is that every median divides the triangle into two smaller triangles, which have exactly equal area. Also, the centroid divides every median into two segments, which are always in the ratio 2:1, from the direction of the vertex.

There is an established formula in geometry, for calculating the length of a median, from the knowledge of the length of a triangle's sides. The length of the median of a triangle named MNO, with sides of lengths m,n, and o is:

L_m = √[(2n² + 2o² - m²) / 4]

Here L_m is the median connecting with a side that has length m. Using this formula, you can calculate the length, without the need to actually draw it. This formula can be derived from what is known as the Apollonius' theorem.