Triangle Inequality theorem

The triangle inequality theorem says that:

In any triangle, the length of a side should be less than the sum and greater than the difference of the lengths of the other two sides

Simply put, the length of a side should lie between the difference and the sum of the other two sides.

The triangle inequality theorem of sides can be written for all three sides as follows

b – c < a < b + c
c – a < b < c + a
a – b < c < a + b

Example:

Can a triangle be formed by sides whose lengths are 3, 4 and 5?

Answer:

Check the triangle inequality property of sides on ‘5’ as follows

Is 4 – 3 < 5 < 4 + 3,

Is 1 < 5< 7?

Yes, it is.

Therefore triangle inequality theorem confirms formation of a triangle if the lengths of the sides are 3, 4 and 5.

Important Tip:

It is sufficient to check whether the triangle inequality property holds on largest length, here 5.

If it does, then on other lengths, it will certainly do; needs no verification.

Example 2:

Can a triangle be formed if the sides are of lengths 5, 10 and 15?

Solution:

Check triangle inequality property on 15, the largest number.

Is 10 – 5 < 15 < 10 + 5

Is 5 < 15 < 15?

Though

5 < 15 is true, but 15 < 15 is false.

Since triangle inequality theorem does not hold, therefore a triangle cannot be formed if the sides have lengths 5, 10 and 15.

The length of the third side cannot be as much as 15, if the other two sides are 5 and 10.

Example 3:

Two sides in an isosceles triangle are 3 and 7. What is the length of the third side?

Solution:

Since the triangle is isosceles, therefore any two sides must be equal in length.

Therefore the third side could be either 3 or 7

Therefore, for the triangle to be isosceles:

the side-triplets must be 3, 3, 7 or 3, 7, 7

Apply triangle inequality thoerem to find out the length of the third side
With the triplet 3, 3 and 7, it is

3 – 3 < 7 < 3 + 3,

0 < 7 < 6, which is not correct.

With the other triplet 3, 7, 7, the triangle inequality property is
7 – 3 < 7 < 7 + 3, i.e.

4 < 7 < 10, which is right.

Therefore, for the triangle to form, the third side has to be of length 7

Short-cut:

Take third side as c

Now, 7 – 3 < c < 7 + 3

4 < c < 10

Between 3 and 7, it is 7 that lies between 4 and 10.

Therefore, third side should be 7.