The triangle inequality theorem is one of the most important geometric theorems in mathematics. It states that the sum of any two sides of a triangle must be greater than the third side. This means that if you have three points on a line, you can use their Cartesian coordinates to find out whether or not it’s possible for them to form a triangle with those coordinates.

## Definition

The **triangle inequality theorem **is a geometric theorem that states that the sum of any two sides of a triangle must be greater than the third side. This means that the shortest side of a triangle will be between the other two sides, not outside of them.

The triangle inequality theorem is used to determine whether or not a given triangle can exist because it requires three vertices and three edges to form a closed object; this requirement prevents many triangles from forming in nature (for example, if we were looking at our environment, there are no perfect equilateral triangles). In addition to its use in determining what types of triangles exist in nature, this statement can also be used when analyzing data sets as part of statistical analysis. For instance, if you were investigating how much time people spend sleeping each night by using an online survey with 100 participants who answered questions based on their own experiences with sleeping habits over seven days (and you wanted to include all possible combinations), then you would need more than four responses per day per person! To do so would violate one or more constraints set forth by this statement–that said: there are other ways around these restrictions without violating anything.

## Sides of a triangle rule: the triangle inequality theorem

One of the most important geometric theorems, the triangle inequality theorem is used to determine whether a given triangle can exist. The theorem can be used in many applications ranging from geometry, trigonometry, and algebra to computer science, quantum physics, and statistics.

The proof of this theorem will be covered in more detail later on in this article.

## Example: using the triangle inequality theorem calculator

The triangle inequality theorem is used in many applications ranging from geometry, trigonometry, and algebra to computer science, quantum physics, and statistics. For example, it is used in geometry to prove that the sum of the lengths of any two sides of any triangle must be greater than the length of the third side. The triangle inequality theorem also has applications in physics where it can be used to determine how much energy a photon has when it travels through space. In trigonometry – we use this theorem when solving problems like these:

- If x + y = 25 then what is x2 + y2?

- If sin(x) = cos(y) then what is tan(x/3)?

- For which values of x does tan(x/2) = 4?

## Minkowski inequality and Hölder inequality

Minkowski inequality is named after Hermann Minkowski, a German mathematician. It is a relation between the sum of two distances and the distance between two points.

The Hölder inequality is an inequality that is used to prove the Minkowski inequality. The Hölder inequality is named after German mathematician Otto Hölder, who proved it in 1907. If you want to learn more about the Minkowski and Hölder inequalities, check out our article on how they’re used in mathematics.

The Triangle Inequality Theorem says that the length of any two sides of a triangle must be greater than or equal to the third side. In other words, if the length of one side is x and the length of another side is y, then there is no way that both x and y could be less than or equal to each other (the same goes for all three sides).

This theorem can be used in many different ways when trying to solve problems involving polygons. You can use it to find relationships between distances and lengths in various polygons.

## FAQ

**What are the three theorems about triangle inequalities?**

The 3 properties of the triangle inequality theorem are: If the sum of any two sides is greater than the third, then the difference between any two sides will be less than the third.

**What is the 3 4 5 Triangle rule?**

This rule says that if one side of a triangle measures 3 and the adjacent side measures 4, then the diagonal between those two points must measure 5 in order for it to be a right triangle.

**Can a triangle be formed with side lengths 4/7 10?**

No. The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.