# Median of triangle

In this chapter, we will learn about the concept of median of triangles along with important properties.

At the end of the chapter, we will solve some problems for your practice.

## What is median of triangle ?

The line segment which start from vertex and meet opposite side of the triangle at midpoint is called median of triangle.

Since any given triangle has three vertex, so we have three medians in a triangle.

Consider the triangle ABC below.

Here line segment AM is the median since it start from vertex A and meet opposite side BC exactly at midpoint M.

Since M is the midpoint, it means that the median divided the opposite side into two equal parts.

Hence, BM = MC

Given below is the image of triangle with all its three medians.

In the given triangle ABC, line segments AM, BN & CL are the three medians.

Since all the three line segments are medians, they divide their respective sides into equal measure.

Median AM divides BC into equal lengths.
BM = MC

Median CL divides AB into equal parts.
AL = LB

Similarly, Median BN divides AC into equal parts
CN = NA

I hope you understood the basic concept of medians. Given below are some important properties that you need to remember.

## Properties of Median

Some important properties of median are;

(a) Median are line segments joining the vertex and mid point of opposite side of the triangle.

(b) Any given triangle has only three medians

(c) The median divide the triangle into two halves with equal area.

Given below is the triangle ABC with median AD.

Since median divide the triangle into equal halves, both the triangles ABD and ACD have equal area.

\mathtt{area\ \triangle ABD\ =\ \triangle ACD}

(d) All the median intersect at a point called centroid of triangle.

In the above triangle, the medians AM, BN & CL meet at point O.

The point O is called centroid.

Centroid is basically center of mass of the given triangle. It is the point across which all the mass of triangle is equally distributed.

(e) The centroid divide each median in the ratio 2 : 1 from the vertex side.

(f) Formula for length of median

For given triangle ABC, let a, b and c are the length of respective sides.

The length of median from vertex A is given as;

\mathtt{m_{a} =\sqrt{\frac{2b^{2} +2c^{2} -a^{2}}{4}}}

Similarly, the length of other two median is given as;

\mathtt{m_{b} =\sqrt{\frac{2c^{2} +2a^{2} -b^{2}}{4}}}\\\ \\ \mathtt{m_{c} =\sqrt{\frac{2a^{2} +2b^{2} -c^{2}}{4}}}

All the formulas are important. Make sure you remember each of them for examination.

(g) Relation between length of median and triangle sides

Note that the sum of all median is always less than the sum of all sides of triangle.

We can say that;

\mathtt{m_{a} +m_{b} +m_{c} < a+b+c}

(h) Sum of median formula

The sum of square of all median is equal to three-forth of square of all sides of triangle.

In the above triangle ABC, the sum of median formula is shown as;

\mathtt{m_{a}^{2} +m_{b}^{2} +m_{c}^{2} =\frac{3}{4}\left( a^{2} +b^{2} +c^{2}\right)}

## Questions on Median of Triangle

Given below are some important solved problems on median of triangle.

First try to solve on your own and then move on to check the given solution.

Example 01
In the below triangle ABC, point O is the centroid of triangle. If length of CD = 4 cm then find length of BD.

Solution
Since O is the centroid of triangle, it means that line segment AD is the median.

We know that median divide the side of triangle into two equal halves.

So, BD = DC

Hence, BD = 4 cm

Example 02
In below triangle ABC, O is the centroid of triangle. If AC = 15 cm, find the length of side AE.

Solution
Since O is the centroid of triangle, the line passing through it is the median.

Here BE is the median of triangle and we know that median divides the sides into equal parts, we can write;

AE = EC

It’s given that AE = 15 cm

AE + EC = 15 cm

2AE = 15 cm

AE = 15/2

AE = 7.5 cm

Hence, the side AE = 7.5 cm

Example 03
Given below is the triangle ABC with AM as a median and O as centroid. If length of AO = 6 cm, find the length of OM.

Solution
We know that centroid divides the median in ration 2 : 1 from the vertex.

So, AO : OM = 2 : 1.

We can also write;

\mathtt{\frac{AO}{OM} =\frac{2}{1}}

Putting the values in above expression, we get;

\mathtt{\frac{6}{OM} =\frac{2}{1}}\\\ \\ \mathtt{OM\ =\ \frac{6}{2}}\\\ \\ \mathtt{OM\ =\ 3\ cm\ }

Hence, length of OM = 6 cm.

Example 04
Given below is triangle ABC with BQ as median and point O as centroid. If median BQ = 24 cm, find the length of BO.

We know that centroid divides the median in ratio 2 : 1 from vertex.

\mathtt{\frac{BO}{OQ} =\frac{2}{1}}\\\ \\ \mathtt{BO\ =\ 2\ .\ OQ}

For the median BQ, we can write;

BO + OQ = BQ

2 (OQ) + OQ = 24

3 (OQ) = 24

OQ = 24 / 3

OQ = 8 cm

We know that BO = 2 (OQ), hence, length of BO is 16 cm.

Example 05
Given below is triangle ABC with median CD and centroid O. If area of triangle CBD = 15 sq. cm. Find the area of triangle ABC.

Solution
We know that median divides the triangle into two equal halves.

\mathtt{Area\ \triangle ABC\ =\ 2\ \times \ \triangle CBD}\\\ \\ \mathtt{\triangle ABC\ =\ 2\ \times \ 15}\\\ \\ \mathtt{\triangle ABC\ =\ 30\ cm^{2}}

Hence, area of triangle ABC is 30 sq. cm.

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