Triangles are **polygons** that are made up of **three sides,** so we can say that the triangle is a flat figure that is made up of **three** different **segments** .

## What is the right triangle?

The right triangle is the triangle that has a **right angle** which has a measure of **90 degrees** and two angles that are **acute** , which means that they measure less than ninety degrees.

- Characteristics of the right triangle
- Properties
- Types of right triangle
- Elements
- Height
- Area
- Perimeter
- Trigonometric ratios
- How to calculate the angle of a right triangle
- Examples

## Characteristics of the right triangle

The most important characteristics of right triangles are:

- They are
**polygons**that have three sides. - They are made up of
**three segments**. - They all have a
**right angle**of ninety degrees. - The two angles that are not right will always be
**acute angles**. - The right angle is made up of the two
**shorter**sides . - They can be
**isosceles**or**scalene**triangles .

## Properties

Right triangles have several properties mentioned below:

- They all have two
**acute angles**and their**hypotenuse**is larger than the legs. - The
**square**of the**hypotenuse**is equal to the**sum**of the square of the**legs**. - The sum of the hypotenuse and the diameter of a circle inscribed in the triangle is equal to the
**sum of the legs**. - When we calculate the area, one leg can be considered as the
**base**and the other leg as the**height**. - The
**median**of the hypotenuse decomposes a scalene right triangle into two triangles. - The median hypotenuse of an isosceles right triangle decomposes it into two congruent and equivalent
**isosceles right**triangles . - Two right triangles that have a
**common hypotenuse**, and the**right angles**in opposite half planes determined by the line containing the hypotenuse, form a**bi-rectangle quadrilateral**. - The median starting from the right angle is equal to
**half the hypotenuse**.

## Types of right triangle

There are two different types of right triangles, these are:

**Isosceles right triangle**: it is the triangle that has a 90º right angle and two 45º angles. The two legs are the same.**Scalene right triangle**: it is the right triangle that has all the different angles always having one of them of 90º. The sides are also different.

## Elements

The elements of the right triangle are:

**Legs**: they are the sides of the triangle that together form the**right angle**.**Hypotenuse**: it is the largest side having the triangle**opposite**the angle**right**.**Right angle**: it is a 90º angle that is formed by the two**legs**.**Acute angles**: are the other two angles of the triangle (α and β) less than 90º. The sum of the two acute angles is**90º.**

## Height

The height of a right triangle can be found using the following theorems:

**Height theorem** : In any right triangle, the height relative to the **hypotenuse** is the geometric mean between the **orthogonal projections** of the **legs** on the hypotenuse.

**h / m = m / h**

If we multiply the two members of the equality by hn then we can obtain: h ^{2} = mn, so that **h = √ (mn)**

## Area

The area that has a right triangle will always have a right angle of **90 °,** so its height must coincide with one of its sides (a). The area of a right triangle is **half** the **product** of the two **sides** that form the right angle (legs a and b). The formula to calculate the area of the triangle is as follows:

**Area = (b a) / 2**

Where b is the **base** and the **side** that coincides with the height.

## Perimeter

We know how the perimeter of a right triangle is the **sum** of the three **sides** . To find out, we apply a simple formula, which is the following:

*Perimeter = a + b + c*

Where a, b and c represent the measures of the triangle.

In addition, the right triangle also complies with the **Pythagorean**** theorem** , so the hypotenuse (c) can be expressed from the legs (a and b). The formula for this type of operation is as follows:

**Perimeter = a + b √ (a² + b²)**

Where a and b are the legs that form the right angle.

## Trigonometric ratios

Right triangles have different **trigonometric ratios** which help us to know the **relationship** between the **sides** and the **angles** of the triangle. Its main function is to show us how much the **internal angles** of the triangle measure when we know the lengths of two sides of the triangle. It is important to remember that right triangles always have a **90 °** angle . There are three common trigonometric ratios which are:

**Sine**(sin): is the ratio that exists between the opposite leg and the hypotenuse and each of them corresponds to an angle.**Cosine**(cos): is the result of dividing the adjacent side by the hypotenuse.**Tangent**(tan): result of dividing the lengths of the opposite and adjacent sides at angle α.

Therefore, we have to:

- Opposite Side / Hypotenuse =
**Sine of the angle** - Adjacent Side / Hypotenuse =
**Cosine of Angle** - Opposite Side / Adjacent Side =
**Tangent of the angle**

## How to calculate the angle of a right triangle

The **heights** of the right triangles are associated with the **legs** (a and b). Therefore, **ha = b** and **hb = a** . The height associated with the **hypotenuse** is known as hc. The three heights of the triangle converge at the **orthocenter** , H at the vertex C of the right angle.

In order to calculate the height associated with side **c (the hypotenuse) ****, the height theorem** is used .

The **height h** can be obtained knowing the three sides of the right triangle and the following formula is applied:

**H = (a b) / ****c**

The right triangle has a **right angle** of 90 °, so its **height** agrees with one of its **sides (a)** . Its **area** will then be **half** the product of the two **sides** that form the **right angle (legs a and b)** . The following formula is used:

**Area = (b a) / ****2**

## Examples

Some examples of how to solve the areas of right triangles are as follows:

#### Let be a right triangle with the sides that form the right angle (a and b) known, where a = 3 cm and b = 4 cm. What is its area?

We apply the previous formula for the area of a right triangle **(b · a) / 2** and it is obtained that its area is **6 cm²** .

#### The area of a right triangle can be obtained from the hypotenuse and the height of the triangle associated with it using the height theorem.

Let n and m be the projections of the legs (bya). Then the area of a right triangle is defined by the following formula:

**Area = (c √ (n m)) / 2**

Formula for the area of a right triangle by the height theorem. This method is useful if the legs (a and b) are not known.