In order to talk about the **hypotenuse** , we must also talk about the **triangles** . The triangle is considered as a **polygon** that has three sides which at the same time give rise to three **vertices** and three internal **angles** . It can be said that the triangles is the simplest figure, after the line in the area of geometry. As a general rule, a triangle is represented by three **capital letters** of the **vertices** (ABC). Triangles can be classified depending on the **measure of their angles** , and in this way we can find acute angles, where all three angles are acute; **right triangle**where the triangle has a right and obtuse angle, with angles greater than 90 °. In the case of the right triangle, the **sides** that form the right angle are called the **legs** and the **opposite side** is called the **hypotenuse** .

## What is the hypotenuse?

The hypotenuse of a triangle is the **side ****opposite** the **right angle** that we find inside a **right triangle** . It can also be said that it is the **longest** side that a right triangle has.

- Definition of hypotenuse
- Etymology
- Properties
- Trigonometric ratios
- How the hypotenuse is calculated
- Examples

## Definition of hypotenuse

The term **hypotenuse** refers to the longest side that we can find in a **right ****triangle** and that has many important properties for both **geometry** and **trigonometry** . In geometry, a hypotenuse is the **longest side** that a right triangle has . We can also say that it is the side **opposite** the **right**** angle** . The word hypotenuse means ** “length below”** or

**” and the origins of the uses and the principle of hypotenuse was thought by Plato in about 400 BC.**

*“stretch below*The hypotenuse of a triangle will always be the **side opposite** the right angle, it is in other words the largest side that a **right triangle** can have . Aside from this side, the other two are known name **opposite side** and **adjacent side** and these names are given because of the relationship with respect to the angle.

## Etymology

The word hypotenuse comes from the Greek word **ὑποτείνουσα** , which is a word formed by a combination of the terms * “hiccup”* which has the meaning

**below**and ”

**teinein**” which means to

**lengthen**. There are some authors who also suggest that the original meaning came from the

**Greek**language and that it arose due to an object that supported something, or from the combination of hiccup that means ”

**below**” and tenuse that had the meaning ”

**side**“.

## Properties

The properties of the hypotenuse are based on the **Pythagorean**** theorem** , which states that the square of the length of the **hypotenuse** is equal to the sum of the squares of the **lengths of** the **legs** . In this way, the length of the hypotenuse is equal to the sum of the lengths of the **orthogonal ****projections** of both legs. These orthogonal projections are as follows:

The **length of** the **hypotenuse** will always be equal to the sum of the lengths of the **orthogonal projections** presented by the two **legs** .

The **square** of the **length of** a leg is equal to the product of the length of its orthogonal projection on the **hypotenuse** and its length, which is equal to · b² = a · mc² = a · n.

Also, the **length of** a leg b is the **proportional mean** that exists between the lengths of its **projection** m and that of the **hypotenuse** a, in other words a / b = b / ma / c = c / n.

## Trigonometric ratios

Remember that the concept of trigonometric ratio refers to the **links** that can be established between the **sides of** a **triangle** which has an angle of 90º. There are three great trigonometric ratios that are: **tangent, sine and cosine** .

In every right triangle that has measurements on its sides independently, we refer to the leg and the hypotenuse, there are also relationships that exist between their sides and that depend on the value of the acute angles of the triangles.

## How the hypotenuse is calculated

The measure of the hypotenuse can be found using the **Pythagorean theorem** , if the **length** of the other two **sides** , called **legs, is known** . According to the Pythagorean Theorem:

- The hypotenuse of a right triangle is equal to the
**square root**of the sum of the square of the**legs**. - The
**leg**of a right triangle is equal to the**square root**of the**hypotenuse**squared minus the other leg squared.

The formula used to calculate the hypotenuse is very simple and we tell you below: **c² = a² + b²** and the result obtained from this equation must be taken as the square root, thus obtaining the hypotenuse.

## Examples

#### Example 1

**Calculate the length of the hypotenuse of a right triangle with sides 1.**

By simple application of the theorem we have that the hypotenuse is:

- H
^{2}= 1^{2 +}1^{2} - H
^{2}= 2 - H = √2

#### Example 2

**In a right triangle, the hypotenuse is 2 and one of its legs is 1. Calculate how long the other leg is.**

- 2
^{2}= 1^{2}+ b^{2} - 4 = 1 + b
^{2} - B
^{2}= 3 - B = √3