Lesson Explainer: Triangle Inequality

In this explainer, we will learn how to define the triangle inequality and identify whether the given side lengths are valid for constructing a triangle.

We can compare the sizes of any relatable quantities that we can represent with numbers using inequalities. Using inequalities in this way allows us to solve real-world problems where we need to compare the sizes or amounts of things. For example, a company may wish to know the shortest distance between two places to save on fuel and travel time.

This idea of the shortest distance combined with inequalities gives motivation to an incredibly useful inequality known as the triangle inequality. To introduce this concept, we start with the idea that the shortest distance between any two points is a straight line. This is a fundamental result in geometry that we use in everyday life even without thinking about it.

For example, if we want to get a drink from across the room, we will instinctively travel in a straight line since this is the shortest distance. Another way of thinking about this being the shortest distance is to say all other paths are the same distance or longer. In particular, this means that any path between two points that travels via another point must be at least as long as the direct path. One way of visualizing this is to think of traveling between two places, say London and Cairo. The shortest path is to travel directly in a straight line, and it would take longer to travel via Moscow.

Ignoring the locations for the moment, we can think of this result entirely in terms of the triangle. The shortest distance between the points is the straight line of length and the other path of length must be longer since it is not direct. Hence,

Since the above journey forms a triangle, we know that the inequality is strict (i.e., is strictly less than ). However, not all choices of journeys form a triangle in this way. For example, a straight line from London to Cairo travels over Athens. If we instead traveled via Athens, our paths would look like the following.

Both paths are direct straight lines, so both paths are equally as long. We have

These two situations showcase a useful result: any side in a triangle must be shorter than the sum of the other two sides. If the length of a side is equal to the sum of the other two sides, then the vertices must be colinear, meaning that the three sides form a line segment rather than a triangle. This is called the triangle inequality, and we can state it formally as follows.

Theorem: The Triangle Inequality

In any triangle , the sum of the lengths of any two sides is greater than the length of the third side.

This means

We can use the triangle inequality to determine if it is possible to construct a triangle with three given side lengths as we will see in our first example.

Example 1: Using the Triangle Inequality to Determine Whether a Triangle Is Possible

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

Answer

We recall that the triangle inequality tells us that the sum of the lengths of any two sides must be greater than the length of the third side. If we sum the lengths of the two shortest lengths, we get

This is shorter than the third length, so these lengths do not satisfy the triangle inequality. Hence, we cannot construct a triangle with these lengths.

Using the triangle inequality, we can show when it is not possible to construct a triangle of given side lengths. However, we can also ask the reverse question: if three side lengths satisfy the triangle inequality, will they always form a triangle? As it turns out, the answer to this question is “yes,” and we can prove it as follows.

Say we want to construct a triangle of side lengths , , and and these values satisfy the triangle inequality. We can assume that the side of length is the longest side. Let’s start by drawing a line segment of length and labeling the endpoints of this line segment and .

Suppose that the side of length is connected to point . If we consider all the possible directions this side can be placed, we can see that they form a circle of radius around , as shown below. We then trace a circle of radius centered at , where we note that the circle will intersect since .

Let’s now do the same thing for the side of length by tracing a circle of radius centered at . Once again, we know that , so this circle must intersect .

Since , the circles must overlap, as the sum of their radii is greater than the distance between their centers. In fact, we know they overlap on a point not on since the inequality is strict. We then call a point of intersection between the circles , and we get the following.

We note that if we used the other intersection point of the circle, it would also form a triangle with the same side lengths. We have shown the following result.

Theorem: Construction of Triangles That Satisfy the Triangle Inequality

If , , and are positive real numbers that satisfy the triangle inequality, then there exists a triangle with side lengths , , and .

In our next example, we will use the theorem to determine which of three given choices are possible lengths for the sides of a triangle.

Example 2: Determining the Possible Lengths of a Triangle’s Sides

Which of the following lists could be the lengths of sides of a triangle?

5, 2, 8
5, 3, 8
2, 5, 6

Answer

We begin by recalling that a triplet of positive numbers can only be the side lengths of a triangle if they satisfy the triangle inequality that states that the sum of the lengths of any two sides must be greater than the length of the third side.

Let us consider the longest side in each case since it is the only side that has the possibility of being greater than the two other sides.

In choice , we note that , so the sums of the sides of lengths 5 and 2 is not greater than the length of the remaining side. So, this triplet does not satisfy the triangle inequality, and so they cannot be the side lengths of a triangle.

In choice , we note that . Once again, the sum of the two shortest sides is not greater than the third. This means that triplet does not satisfy the triangle inequality, and so they cannot be the side lengths of a triangle.

Finally, in choice , we see that

All three sums are larger than the length of the third side, so these 3 lengths satisfy the triangle inequality. It is worth noting that we only needed to consider the first inequality since we just need to check if the sum of the two shortest sides is greater than the longest side.

Hence, the triplet 2, 5, 6 can be the lengths of the sides of a triangle.

It is worth noting that we can construct the triangle in the previous example by drawing a line of length 6 cm and then tracing arcs of circles of radii 5 cm and 2 cm centered at the end points of the line to get the following.

In our next example, we will find the range of all possible values for an unknown in a missing side length of a triangle using the triangle inequality.

Example 3: Finding the Range of All Possible Values of a Triangle’s Side Length given the Lengths of the Other Sides

Find the range of all possible values of if cm, 2 cm, and 25 cm represent the lengths of the sides of a triangle.

Answer

We first recall that the side lengths in a triangle must be positive and satisfy the triangle inequality. We also recall that if a triplet of three numbers satisfies the triangle inequality, then these values can represent the side lengths of a triangle.

The triangle inequality states that the sum of the lengths of any two sides must be greater than the length of the third side. We can use this to construct three inequalities that the side lengths must satisfy for the values to represent the lengths of the sides of a triangle. It is worth noting that since , we do not need to consider the inequality for 2 because it will always be smaller than 25 plus the other side length (since it must be positive).

We have

We will solve both inequalities for . Simplifying the first inequality gives

Subtracting 8 from both sides of the inequality yields

Simplifying the second inequality yields

Subtracting 6 from both sides of the inequality gives us

Therefore, we need and . We can write this as the compound inequality .

In our next example, we will combine the triangle inequality with our knowledge of isosceles triangles to determine the missing side length in an isosceles triangle.

Example 4: Finding the Missing Side Length of an Isosceles Triangle

If is an isosceles triangle with and , find .

Answer

We know that an isosceles triangle must have a pair of congruent sides. So, for to be an isosceles triangle, the missing side must be congruent to one of or . This means the possible lengths of are 2 cm and 5 cm.

We then recall that the side lengths in any triangle must satisfy the triangle inequality. The triangle inequality tells us that the sum of the lengths of any two sides must be greater than the length of the third side and if this holds true, then the triangle can be constructed.

Therefore, we can check each of the triplets 2, 2, 5 and 2, 5, 5 to determine if they can represent the side lengths of a triangle.

Consider the first triplet, 2, 2, 5. We see that , so a side of length 5 would be longer than two sides of length 2, which means that this triplet does not satisfy the triangle inequality.

For the second triplet, 2, 5, 5, we see that

It is worth noting that we only needed to consider the first inequality since we just need to check if the sum of the two shortest sides is greater than the longest side.

Hence, all three sums are larger than the length of the third side, which means that this triplet satisfies the triangle inequality.

Therefore, .

In our final example, we will prove a relationship between the side lengths of a triangle and its perimeter using the triangle inequality.

Example 5: Using the Triangle Inequality to Complete a Proof

Consider triangle , where , , and .

Fill in the blanks using , , , , or = in the following statements.

For any side of the triangle, say .
Therefore, we can conclude that .
If we denote the perimeter of the triangle by , then .

Answer

Part 1

The triangle inequality tells us that the sum of the lengths of any two sides must be greater than the length of the third side. This means that the sum of the sides of lengths and must be greater than . Giving us

Hence, the correct symbol is >.

Part 2

If we add to both sides of the inequality in part 1, we get

Hence, the correct symbol is >.

Part 3

We recall that the perimeter of a polygon is equal to the sum of all of its side lengths. So, in a triangle with sides of lengths cm, cm, and cm, if the perimeter is cm, it must be

This is equal to the left-hand side of the previous inequality, so we have

Dividing both sides of the inequality by 2 gives us

Hence, the correct symbol is >.

In the previous example, we proved that half the perimeter of a triangle is greater than any of the side lengths or, alternatively, that the perimeter of a triangle is always greater than twice any of its side lengths.

Let’s finish by recapping some of the important points from this explainer.

Key Points

The triangle inequality tells us that in any triangle , the sum of the lengths of any two sides is greater than the length of the third side.
This means
If the length of one side of a triangle is equal to the sum of the lengths of the other two sides, then the vertices are colinear, and so the shape is not really a triangle but a line.
If , , and are positive real numbers that satisfy the triangle inequality, then there exists a triangle with side lengths , , and .
We can prove some geometric relationships using the triangle inequality. For example, the perimeter of any triangle is always greater than twice any of its side lengths.