In this explainer, we will learn how to apply the triangle inequality to determine whether three given lengths can be sides of a triangle.

### The Triangle Inequality

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

This leads to three inequalities.

When constructing a triangle with a ruler and a pair of compasses, you need to draw the base of the triangle first. Then, you draw a circle centered at one of the baseβs endpoints of a radius the length of the second side, and finally a second circle centered at the other baseβs endpoint of a radius the length of the third side. The third vertex of the triangle is then given by the intersection of both circles, as shown in the diagram.

### How to Understand the Triangle Inequality

Letβs try to construct triangles of sides 7 cm, 4 cm, and 2 cm. By drawing the 7 cm long side first and then the two circles of radii 4 cm and 2 cm, we get the following diagram.

The two circles do not intersect because . This means that a
triangle of sides 7 cm,
4 cm, and
2 cm does **not** exist.

Note that it does not matter which side you draw as a base, as shown here. Of course, you still cannot construct any triangle.

### Example 1: Using the Triangle Inequality

Is it possible to form a triangle with side lengths 3 inches, 5 inches, and 7 inches?

### Answer

We need to check that each of the triangle inequalities is true with these side lengths.

- Is ? Yes.
- Is ? Yes.
- Is ? Yes.

Therefore, it is possible to form a triangle with side lengths 3 inches, 5 inches, and 7 inches.

### Example 2: Using the Triangle Inequality

Is it possible to form a triangle with side lengths 6 m, 7 m, and 18 m?

### Answer

We need to check that each of the triangle inequalities is true with these side lengths.

- Is ? No.

Therefore, it is **not** possible to form a triangle with side lengths
6 m,
7 m, and
18 m.

Note that as soon as we find that one of the triangle inequalities is not true, we know that it is not possible to form the triangle. We do not need to check the other triangle inequalities.

### Example 3: Using the Triangle Inequality

Two sides of a triangle are 5 cm and 8 cm. What is the range of values for the third side?

### Answer

We are given two side lengths of a triangle, and we are asked about the range of values for the third sideβs length. Let the length of the third side be cm. Letβs now use the triangle inequalities to find the range of values for .

- . Taking away 5 from each side, we get .
- . Taking away 8 from each side, we get . This does not give us any new information as is a length in centimeters and is thus positive. Furthermore, we know from (1) that .
- ; that is , or .

By combining (1) and (3), we get .

### Example 4: Using the Triangle Inequality

Given that is an integer, determine the possible values of .

### Answer

We are going to use the three triangle inequalities to find the range of values for .

- ; by adding the like terms, we get ; by adding 3 to each side, we get ; and by taking away from each side, we get , or .
- ; by adding the like terms, we get ; by subtracting from each side, we get ; and by dividing each side by 3, we get .
- ; by adding the like terms, we get ; by subtracting from each side, we get ; by adding 4 to each side, we get ; and by dividing each side by 7, we get .

By combining (1) and (3), we get . As we are told that is an integer, we find that .

Note that (2) did not give us any valuable information as is a length and is thus positive, and (3) gave that .