2.4 Triangle Inequalities

Triangle Inequality Theorem

The triangle inequality theorem states that, in a triangle, the sum of lengths of any two sides is greater than the length of the third side. Suppose a, b and c are the lengths of the sides of a triangle, then, the sum of lengths of a and b is greater than the length c. Similarly, b + c > a, and a+ c > b. If, in any case, the given side lengths are not able to satisfy these conditions, it means it is not possible to draw a triangle with those measurements.

What is Triangle Inequality Theorem?

The triangle inequality theorem states, “The sum of any two sides of a triangle is greater than its third side.” This theorem helps us to identify whether it is possible to draw a triangle with the given measurements or not without actually doing the construction. Let’s understand this with the help of an example. Triangle ABC has side lengths of 6 units, 8 units, and 12 units.

Here, AB = 6 units, BC = 8 units and CA = 12 units.

The sum of sides AB and BC is 6 + 8 = 14 units and 14 is greater than side CA (12 units).
The sum of sides BC and CA is 8 + 12 = 20 units and 20 is greater than side AB (6 units).
The sum of sides CA and AB that is 12 + 6 = 18 units and 18 is greater than side BC (8 units).

Thus, lengths of all the sides satisfy the triangle inequality theorem. In this, not only one, but all 3 cases should satisfy the triangle inequality theorem.

Let’s take another example. Let’s check whether a triangle with sides lengths 5 units, 3 units, and 10 units satisfy the triangle inequality theorem or not.

Here,

5 + 3 = 8 which is less than 10
3 + 10 = 13 which is greater than 5
10 + 5 = 15 which is greater than 3

We can see that two cases are satisfying the triangle inequality theorem but one case is not satisfying. This means the triangle with these side lengths does not exist. All three sides should satisfy the triangle inequality theorem.

Triangle Inequality Theorem Formula

Before understanding the formula, first, we need to understand the proof of the triangle inequality theorem. Consider a triangle ABC as shown below.

Let us extend side AB to the point D such that AC = AD and △BDC will form a right angled triangle at angle C.

Applying angle sum property in △BDC, we get,

∠BDC + ∠CBD + ∠BCD = 180°

∠BDC + ∠CBD + 90° = 180°

∠BDC + ∠CBD = 90°

This implies, ∠BCD > ∠BDC.

As the side opposite to the greater angle is longer, we have BD > BC.

This implies:

BD > BC

AB + AD > BC

AB + AC > BC

Hence proved.

Similarly, we can prove that AC + BC > AB and AB + BC > AC.

So, the triangle inequality theorem formula is,

AB + AC > BC
AC + BC > AB
AB + BC > AC

Example 1: Suzie has three sticks of lengths 4 units, 8 units, and 2 units. Using the triangle inequality theorem, find out whether Suzie can form a triangle using these sticks or not?

Solution: The triangle formed by the given sticks must satisfy the triangle inequality theorem.

Let’s check if the sum of the two sides is greater than the third side.

4 + 8 > 2 ⟹ 12 > 2 ⟹ TRUE
2 + 8 > 4 ⟹ 10 > 4 ⟹ TRUE
4 + 2 > 8 ⟹ 6 > 8 ⟹ FALSE

So, the lengths of the sticks do not satisfy the triangle inequality theorem. Thus, Suzie can’t form a triangle using the sticks of the given lengths.

Example 2: Ron wants to decorate his triangular flag with a ribbon. The two sides of the flag are 8 units and 2 units. Using the triangle inequality theorem, find out how much ribbon is required for the third side?

Solution: By using the triangle inequality theorem, we can say that the length of the third side must be less than the sum of the other two sides.

So, the third side is less than 8 units + 2 units = 10 units.

Also, the third side cannot be less than the difference between the other two sides.

So, the third side is greater than 8 units – 2 units= 6 units.

Thus, the length of the ribbon can be 7, 8, or 9 units.