Median of Triangle: Definition and Essential Properties

median of triangleCalculating the median of a triangle is one of the fundamental problems in geometry. This tutorial will teach you what the median is, how to calculate it, and how to solve problems relating to it.

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Different Types of Triangles

Before we can define the median of a triangle, we must first learn about the different types of triangles. This will come in handy when we are working with medians.

Depending on the number of equal sides, triangles may be classified as:

  • Scalene: A triangle with no equal sides or angles, i.e. all three sides of the triangle are different. In triangle ABC given below, all three sides are of different lengths.

scalene

  • Isosceles: A triangle with at two equal sides. In triangle ABC given below, sides AB and AC are equal.

scalene

  • Equilateral: A triangle where all sides are equal. In triangle ABC shown below, sides AB = BC = CA.

equilateral

Remember that if the sides of a triangle are equal, the angles opposite the side are equal as well. For example, in equilateral triangle ABC shown above, since AB = BC = CA, ∠ACB = ∠BAC = ∠ABC.

With that out of the way, we can learn more about medians in a triangle.

What is the Median of a Triangle?

A median of a triangle is the line segment that joins any vertex of the triangle with the mid-point of its opposite side. In the figure shown below, the median from A meets the mid-point of the opposite side, BC, at point D.

median of a triangle

Hence, AD is the median of ∆ABC and it bisects the side BC into two halves where BD = BC.

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Properties of Median of a Triangle

A median has some peculiar characteristics, such as:

1. In isosceles and equilateral triangles, the median bisects the angle at the vertex whose two adjacent sides are equal.

The median not only bisects the side opposite the vertex, it also bisects the angle of the vertex in case of equilateral and isosceles triangles, provided the adjacent sides are equal as well (which is always true in case of equilateral triangles). In equilateral triangle ABC shown below, median AD bisects ∠BAC such that ∠BAD = ∠CAD.

property1

2. A triangle can have only three medians, all of which intersect at a point called ‘centroid’.

Since a triangle has three vertices, it follows that it can have only three medians. Interestingly, all medians of a triangle intersect at a single point called the centroid. It doesn’t matter what the shape or size of the triangle, the medians will always intersect at the centroid.

In ∆ABC shown below, medians AD, BE and CF intersect at point G, which forms the centroid.

g-centroid

3. A median divides the area of the triangle in half.

In any triangle ABC, the median AD divides the triangle into two triangles of equal area.

Here, the total area of ∆ADB = area of ∆ADC.

median of a triangle

4. The centroid divides the length of each median in 2:1 ratio.

The length of the part between the vertex and the centroid is twice the length between the centroid and the mid-point of the opposite side.

For example, in the triangle shown below, length of AG is twice the length of GD, while length of BG is twice the length of GE.

centroid

5. The centroid divides the triangle into six smaller triangles of equal area.

This is another very interesting property of the centroid. As can be seen in the figure above, the centroid basically divides the triangle into six smaller triangles, namely triangles AGE, CEG, CGD, DGB, CGF and FGA.

Interestingly, the area of all these triangles is equal. Thus, the centroid not only divides the medians into 2:1 ratio, but also divides the triangle into six triangles of equal area.

6. The length of medians in an equilateral triangle is always equal.

Since the length of all sides in an equilateral triangle is equal, it follows that the length of medians bisecting these sides is equal as well.

Thus, in equilateral triangle ABC where AD, BE and CF are medians originating from A, B and C respectively, we have:

AD = BE = CF

property6

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7. In an isosceles triangle, medians drawn from vertices with equal angles are equal in length.

Based on the above, it follows that the length of medians originating from vertices with equal angles should be equal.

Thus, in an isosceles triangle ABC where AB = AC, medians BE and CF originating from B and C respectively are equal in length.

property7

8. In a scalene triangle, all medians are of different length.

Based on the above two properties, we can easily conclude that since all sides are unequal in length in a scalene triangle, the medians must also be unequal. Thus, regardless of the shape of the scalene triangle, all of its medians will have different lengths.

9. The length of a median can be calculated using Apollonius Theorem.

Named after Greek astronomer Apollonius of Perga, the Apollonius Theorem is used to calculate the length of a median of a triangle, provided we know the length of its sides.

property9

Here, in triangle ABC:

  • a = Length of side opposite vertex A.
  • b = Length of side opposite vertex B.
  • c = Length of side opposite vertex C.
  • ma = Length of median originating from vertex A.
  • mb = Length of median from vertex B.
  • mc = Length of median from vertex C.

This can be illustrated as below:

property9illustration

For example, if we have a triangle with the following side measurements:

property9example

Here, length of median mcan be calculated as:

√[2(42) + 2(62) – 82]/4

= √40/4

= √10

= 3.162

This concludes our tutorial on medians of a triangle. We’ve learned a number of interesting properties of medians, including how they divide the triangle into two equal halves, intersect at the centroid and bisect the opposite side. To learn more advanced geometry concepts, try your hand at this course on introductory geometry.