Mathematics# Right triangle

## What is a right triangle?

## Characteristics of the right triangle

## Properties

## Right triangle types

## Elements

## Height

## Area

## Perimeter

## Trigonometric ratios

## How to calculate the angle of a right triangle

## Examples

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

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

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

**Related topics**

Cathetus, hypotenuse, Pythagoras, Pythagorean theorem, isosceles triangle, scalene 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.

The most important characteristics of right triangles are:

- They are
**polygons**that have three sides. - They are formed by
**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 side** - They can be
**isosceles**or**scalene**

Right triangles have several properties that are mentioned below:

- They all have two
**acute angles**and their**hypotenuse**is larger than the cathetus. - The
**square**of the**hypotenuse**is equal to the**sum**of the square of the**cathetus**. - The sum of the hypotenuse and the diameter of a circle inscribed in the triangle is equal to the
**sum of the cathetus**. - When calculating the area, one cathetus can be considered the
**base**and the other one as the**height**. - The
**median**of the hypotenuse breaks down a right-scaled scalene triangle into two triangles. - The median of the hypotenuse of an isosceles right triangle breaks it down 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**birectangular****quadrilateral**. - The median that starts from the right angle is equal to
**half of the hypotenuse**.

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

**Right isosceles triangle:**it is the triangle that has a right angle of 90º and two angles of 45º. The two legs are equal.**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.

The elements of the right triangle are:

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

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 median between the **orthogonal projections** of the **cathetus** over the hypotenuse.

**h/m = m/h**

If we multiply the two equality members by hn then we can get: h2 = mn, so **h = √(mn)**

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 (cathetus a and b). The formula for calculating the area of the triangle is as follows:

**Area = ( b – a ) / 2**

Being b the **base** and the **side** that matches the height.

We know as the perimeter of a right triangle 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 measurements of the triangle.

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

**Perimeter = a + b √(a²+b²)**

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

Right triangles have different **trigonometric ratios** which help us know the **relationship** between the **sides** and **angles** of the triangle. Their 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 reasons that are:

**Sinus**: is the reason that exists between the opposite cathetus 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 to the angle α.

Therefore, we have to:

- Opposite Side / Hypotenuse =
**Angle Sine.** - Adjacent Side / Hypotenuse =
**Angle Cosine.** - Opposite Side / Adjacent Side =
**Angle Tangent**.

The **heights** of the right triangles are associated with the **cathetus** (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 in 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 by 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** matches one of its **sides** (a). Its **area** will then be **half** the product of the two **sides** that form the **right angle (cathetus a and b).** The following formula is used:

**Area = ( b – a ) / 2**

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

We apply the previous formula of the area of a right **triangle ( b – a ) / 2** and we get that its area is **6 cm².**

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

**Area = ( c · √( n·m)) / 2**

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

Written by Gabriela Briceño V.

Briceño V., Gabriela. (2019). *Right triangle*. Recovered on 28 October, 2023, de Euston96: https://www.euston96.com/en/right-triangle/