# Hypotenuse

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 trianglewhere 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 “stretch below ” and the origins of the uses and the principle of hypotenuse was thought by Plato in about 400 BC.

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:

• 2 = 1 2 + 1 2
• 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 = 1 2 + b 2
• 4 = 1 + b 2
• 2 = 3
• B = √3