Maths-

General

Easy

Question

# Prove the following statement.

The length of any one median of a triangle is less than half the perimeter of the triangle.

Hint:

- Triangle inequality theorem
- According to this theorem, in any triangle, sum of two sides is greater than third side,
- a < b + c
- b < a + c
- c < a + b

- The line segment which joins the vertex of triangle to midpoint of opposite side is called the ‘median’.
- Perimeter of triangle is sum of sides.
- Perimeter = a + b + c

- The line segment which joins the vertex of triangle to midpoint of opposite side is called the ‘median’.
- Perimeter of triangle is sum of sides.
- Perimeter = a + b + c

## The correct answer is: Hence, proved.

- The length of any one median of a triangle is less than half the perimeter of the triangle.

- Step 1:

Let,

Side AB = a

Side BC = b

Side AC = c

And median AM be M_{a}

- Step 2:

In △ABM,

AM is median so M is midpoint of BC

BM =

So, according to triangle inequality theorem,

M_{a} < a + ---- eq. 1

In △ACM,

AM is median so M is midpoint of BC

MC =

So, according to triangle inequality theorem,

M_{a} < c + ---- eq. 2

Add eq. 1 and eq. 2.

M_{a} + M_{a} < a + + c +

2M_{a} < a + b + c

- Final Answer:

Hence, proved.

_{a}

_{a}< a + ---- eq. 1

_{a}< c + ---- eq. 2

Add eq. 1 and eq. 2.

_{a}+ M

_{a}< a + + c +

_{a}< a + b + c