Relation between Median and Side of Triangle

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What is the Median?

The mathematical word "median" has different meanings. In statistics, it is the value lying at the center of a data set. So for a data set {3, 5, 7, 9, 11}, 7 is the median. In geometry, a median is a line segment from an angle of a triangle to the midpoint of the opposite side of a triangle. A median may or may not be perpendicular to the side of the triangle. The intersection of the 3 medians is called a centroid.

In the case of some triangles like the equilateral triangle, the median, and altitude are the same.

A triangle has 3 medians. Median can also be defined as the line from the midpoint of a side to the opposite interior angle of the triangle. The medians are concurrent at the centroid. The point which is common to all the 3 medians at their intersection is called a point of concurrency, the centroid of a triangle.

For example, if there is a triangle ABC, the line segment from A meets the mid-point of the opposite side, BC, at point D. Therefore, AD is the median of ∆ABC and it bisects the side BC into two halves where BD = BC.

The relation between the median and the sides of a triangle is such that “3 times the sum of squares of the length of sides = 4 times the squares of medians of a triangle.”

3 (AB2 + BC2 + CA2) = 4 (AD2 + BE2 + CF2)

Properties of Median of a Triangle

A median has some unique properties, and these properties are discussed below.

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

The median not only bisects the side opposite to the vertex, but it also bisects the angle of the vertex in the case of equilateral and isosceles triangles. In the equilateral triangle ABC shown below, median AD bisects ∠BAC such that ∠BAD = ∠CAD.

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

As a triangle has three vertices, so it can have only three medians. All medians of a triangle intersect at a single point called the centroid. The shape or size of the triangle, does not matter, and the medians will always intersect at the centroid.

In ∆ABC, medians AD, BE, and CF intersect at point G, which is called the centroid of the triangle.

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3. A median divides the area of the triangle in two equal halves.

In a triangle ABC, the median AD divides the triangle into two equal triangles whose areas are equal.

Area of ∆ADB = Area of ∆ADC.

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

The length of the part between the vertex and the centroid is twice the length between the centroid and the midpoint of the opposite side.

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

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

From the figure above, the centroid divides the triangle into six smaller triangles, namely triangles AGE, CEG, CGD, DGB, BGF, and FGA. The areas of all these triangles are equal. Thus, the centroid divides the medians into a 2:1 ratio, as well as divides the triangle into six smaller triangles of equal area.

6. The length of the median of an equilateral triangle is always equal.

As we know that the length of all sides of an equilateral triangle is equal, it follows that the length of the median of an equilateral triangle bisecting these sides is also equal.

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

AD = BE = CF

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

The length of medians drawn from vertices with equal angles should be equal.

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

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8. The median and sides of the triangle are related in such a way that “3 times the sum of the squares of the length of sides = 4 times the squares of medians of a triangle.”

3 (AB2 + BC2 + CA2) = 4 (AD2 + BE2 + CF2)


How to Find the Median of a Triangle

A theorem, called Apollonius's Theorem, can help you to find the median of a triangle. 

The formula to find the median of a triangle is given as, 

m= \[\sqrt{\frac{2b^{2}+c^{2}-a^{2}}{4}}\]

where a, b, and c are the lengths of the sides and m is the median from interior angle drawn from the vertex A to side a.


Solved Examples

Example 1: In the adjoining figure given, ∠PQR = 90∘ and QL is a median, PQ = 12cm, and QR = 14cm. Find QL.

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Solution:

We have  PQ = 5 cm, QR = 12 cm and QL is a median.

∴ PL = LR = PR2 …….(I)

In ΔPQR, 

(PR)2 = (PQ)2 + (QR)2 ………...(By Pythagoras theorem)

= 144 + 256 

= 400

⇒ PR = 20

Now, by theorem, if L is the mid-point of the hypotenuse PR of a right angled ΔPQR, then,

QL = 1 / 2 [PR] = [1 / 2] * (20) = 10cm.


Example 2: In a triangle, ABC, coordinates of A are (1, 2) and the equations of the medians through B and C are x + y = 5 and x = 4 respectively, then what is the area of ΔABC (in sq. units)?

Solution:

Median through C is x = 4

So, clearly, the x coordinate of C is 4

Let C = (4, y), then the midpoint of A (1, 2) and C (2, y) which is D lies on the median through B by definition.

Clearly, D = ([1 + 4] / 2, [2 + y] / 2).

Now, we have, [3 + 4 + y] / 2 = 5 ⇒ y = 3. So, C = (4, 3).

The centroid of the triangle is for by the intersection of the medians.

It is easy to see that the medians x =4 and x + y = 5 intersect at G = (4, 1).

The area of triangle ΔABC

 = 3 × ΔAGC 

= 3 × 1 / 2 × 3 × 2 

= 9.