A median of a triangle is a line segment from a vertex to the midpoint of the side opposite that vertex.

**Medians
of a Triangle**

A median
of a triangle is a line segment from a vertex to the midpoint of the side opposite
that vertex.

In the figure is a median of ∆ABC.

Are there
any more medians for ∆ABC ? Yes, since there are three vertices in a triangle, one
can identify three medians in a triangle.

__Example 5.16__

In the figure,
ABC is a triangle and AM is one of its medians. If BM = 3.5 *cm*, find the length of the side BC.

*Solution:*

*AM *is median* *⇒ *M *is the midpoint of* BC*.

Given that,
*BM* = 3.5 *cm*, hence *BC* = twice the length
*BM* = 2 **×** 3.5 *cm* = 7 *cm*.

__Activity__

1. Consider a paper cut-out of a triangle. (Let us have an acute-angled
triangle, to start with). Name it, say ABC.

2. Fold the paper along the line that passes through the point A
and meets the line BC such that point B falls on C. Make a crease and unfold the
sheet.

3. Mark the mid point M of BC.

4. You can now draw the median AM, if you want to see it clearly. (Or you can leave it as a fold).

5. In the same way, fold and draw the other two medians.

6. Do the medians pass through the same point?

Now you can repeat this activity for an obtuse - angled triangle
and a right triangle.

What is the conclusion? We see that,

**The three medians of any
triangle are concurrent.**

__1. Centroid__

The point
of concurrence of the three medians in a triangle is called its **Centroid**, denoted by the letter **G**.Interestingly, it happens to be the centre
of mass of the triangle. One can easily verify this fact. Take a stiﬀ
cut out of triangle of paper. It can
be balanced horizontally at this point on a finger tip or a pencil tip.

Should you
fold all the three medians to find the centroid? Now you can explore among yourself
the following questions:

(i) How can
you find the centroid of a triangle?

(ii) Is the centroid equidistant from the vertices?

(iii) Is the centroid of a triangle always in its interior?

(iv) Is there anything special about the medians of
an (a) Isosceles triangle? (b) Equilateral triangle?

**Properties of the centroid
of a triangle**

The location
of the centroid of a triangle exhibits some nice properties.

• It is always
*located inside the triangle*.

• We have
already seen that it *serves as the Centre
of gravity *for any triangular lamina.

• Observe
the figure given. The lines drawn from each vertex to G form the three triangles
∆ABG, ∆BCG, and ∆CAG.

Surprisingly,
the areas of these triangles are equal.

*That is, the medians of a triangle divide
it into three smaller triangles of equal area!*

*The centroid of a triangle
splits each of the medians in two segments, he one closer to the vertex being twice
as long as the other one.*

This means the centroid divides each median in a ratio of 2:1. (For
example, GD is ⅓ of PD).

*(Try to verify this by
paper folding)*.

__Example 5.17__

In the figure
*G* is the centroid of the triangle XYZ.

(i) If GL
= 2.5 *cm*, find the length XL.

(ii) If YM
= 9.3 *cm*, find the length GM.

*Solution:*

(i) Since
G is the centroid, XG : GL = 2 : 1 which gives XG : 2.5 = 2 : 1.

Therefore,
we get 1 × (XG) = 2 × (2.5)

⇒ XG
= 5 *cm*.

Hence, length
XL = XG + GL = 5 + 2.5 = 7.5 *cm*.

(ii) If YG
is of 2 parts then GM will be 1 part. (Why?)

This means YM has 3 parts.

3 parts is
9.3 *cm* long. So GM (made of 1 part) must
be 9.3 ÷ 3 = 3.1 *cm*.

__Example 5.18__

ABC is a
triangle and G is its centroid. If AD=12 *cm*,
BC=8 *cm* and BE=9 *cm*, find the perimeter of ∆BDG **.**

*Solution:*

ABC is a
triangle and G is its centroid. If,

The perimeter of ∆BDG = BD+GD+BG = 4+4+6 = 14 *cm*

*Solution 2:*

ABC is a triangle and G is its centroid. If,

AD = 12 *cm*

⇒
GD = 1/3 of AD = 1/3(12) = 4 *cm* and BE
= 9 *cm*

⇒ BG
= 2/3 of BE = 2/3 (9) = 6*cm* .

Also D is
a midpoint of BC

⇒
BD = 1/2 of BC = 1/2 (8) = 4*cm*.

The perimeter
of ∆BDG = BD+GD+BG = 4+4+6 = 14 *cm*

Tags : Geometry | Chapter 5 | 8th Maths , 8th Maths : Chapter 5 : Geometry

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8th Maths : Chapter 5 : Geometry : Medians of a Triangle | Geometry | Chapter 5 | 8th Maths

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