# What are the characteristics of right triangles?

A right triangle is one of the most important figures in geometry and forms the basis for trigonometry. These triangles are characterized by having an angle of 90 degrees. Because of this, the right triangles generate the most important theorem which is the Pythagorean theorem.

Here, we will look at a definition of right triangles and we will learn about their most important characteristics. Also, we will learn about its most important formulas and apply them to solve some problems.

##### GEOMETRY

Relevant for

Learning about the characteristics of right triangles.

See characteristics

##### GEOMETRY

Relevant for

Learning about the characteristics of right triangles.

See characteristics

## Definition of a right triangle

A right triangle is a triangle that has an angle of 90 degrees. These triangles have three sides, “base,” “hypotenuse”, and “height”, where the angle between the base and the height is 90 degrees. This triangle is a very important figure in mathematics because it gives rise to the Pythagorean theorem.

Recall that the Pythagorean theorem tells us that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other sides. For example, in the right triangle below, the hypotenuse is “c” since it is the side opposite the 90-degree angle and the other sides have lengths a and b:

## Fundamental characteristics of right triangles

The following are the most important characteristics of right triangles:

• An angle of this triangle is always 90°, that is, a right angle.
• The opposite side of the right angle is the hypotenuse.
• The hypotenuse is always the longest side.
• The sum of the other interior angles is equal to 90°.
• The other two sides adjacent to the right angle are called the base and the perpendicular.
• If we draw a circumcircle that passes through the three vertices, then the radius of this circle is equal to half the hypotenuse.
• If one of the angles is 90° and the other two angles measure 45° each, then the triangle is called an isosceles right triangle, where the sides adjacent to the 90° angle are equal.

## Important right triangle formulas

The three most important formulas for right triangles are the area formula, the perimeter formula, and the Pythagorean theorem.

### Area of a right triangle

The area of a right triangle is calculated using the length of the base and the length of the height:

where b is the length of the base and h is the length of the height.

### Perimeter of a right triangle

The perimeter of a right triangle is calculated by adding the lengths of all the sides:

where, $latex a, ~b, ~ c$ are the lengths of the sides of the triangle.

### Pythagoras theorem

The Pythagorean theorem allows us to find the length of the hypotenuse of the triangle if we know the lengths of the other two sides:

where, $latex a, ~ b$ are the lengths of the sides and c is the length of the hypotenuse.

## Examples of right triangle problems

### EXAMPLE 1

• What is the area of a triangle that has a height of 12 m and a base of 14 m?

Solution: We have $latex b=14$ and $latex h=12$. Therefore, we use the area formula with these values:

$latex A= \frac{b \times h}{2}$

$latex A= \frac{14 \times 12}{2}$

$latex A= \frac{168}{2}$

$latex A=84$

The area of the triangle is 84 m².

### EXAMPLE 2

• What is the perimeter of a triangle that has sides of length 6 m, 8 m, and 10 m?

Solution: We have the lengths $latex a=6$, $latex b=8$ and $latex c=10$. Therefore, we use the perimeter formula with these values:

$latex p= a+b+c$

$latex p= 6+8+10$

$latex p=24$

The perimeter of the triangle is 24 m.

### EXAMPLE 3

• ¿Cuál es la longitud de la hipotenusa de un triángulo rectángulo que tiene lados de longitud 5 m y 12 m?

Solution: We have the lengths $latex a=5$ and $latex b=12$. Therefore, we use the Pythagorean theorem with these values:

$latex {{c}^2}= {{a}^2}+{{b}^2}$

$latex {{c}^2}= {{5}^2}+{{12}^2}$

$latex {{c}^2}= 25+144$

$latex {{c}^2}=169$

$latex c=13$

The length of the hypotenuse is 13 m.