1 December 2023

As a student or former student, it is very likely that you have had to deal with the Pythagorean theorem. This establishes the fundamental principles of mathematics and geometry. The theorem works for all sorts of applications in mathematics. Are you clear about what the Pythagorean theorem is about? Well, let’s start!

## Concept and definition of the Pythagorean theorem

The Pythagorean theorem is a geometric statement. quite recognized within the mathematical world physical, which establishes a close relationship between the sides of a right triangle.

a right triangle It is characterized by having a right angle, that is, 90 degrees. The 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 legs.

In short, if the sides of a right triangle are represented as a, b, and c (with c being the hypotenuse), then the Pythagorean theorem states that c^2 = a^2 + b^2.

This mathematical logic has a large number of applications in various areas of mathematics and physics. For example, it is used to solve trigonometry problems, calculate distances on a Cartesian plane, and in number theory to find integers that satisfy the equation a^2 + b^2 = c^2.

The Pythagorean theorem is named after the Greek mathematician of the same name, who is considered to be the first researcher to prove it rigorously. However, the relationship between the sides of right triangles was known to other cultures, such as the Babylonians and Egyptians, long before Pythagoras formally established it.

## What is the Pythagorean theorem used for?

The Pythagorean theorem is named after in honor of the Greek mathematician Pythagoras. Although it is known that other cultures, such as the Babylonians and Egyptians, already knew the relationship between the sides of right triangles before Pythagoras formalized it.

This theorem has great utility in mathematics and physics, since it allows us to calculate the length of a side of a right triangle as a function of the other two.

For example, when the lengths of two sides of a right triangle are known, the Pythagorean theorem can be applied to determine the length of the hypotenuse, allowing us to calculate the height of a building or the distance between two points on a map.

Furthermore, the Pythagorean theorem also is important in mathematics education, since it is one of the first tools taught in school and is used to introduce students to concepts such as irrational numbers and trigonometric functions. In summary, the Pythagorean theorem is a valuable and essential tool in mathematics and physics, with practical applications in many areas of life.

## What is the history and origin of the Pythagorean theorem?

The history of the Pythagorean Theorem dates back to ancient Mesopotamia and Egypt, where mathematicians already knew some special cases of what we know today as the Pythagorean theorem. For example, in ancient Babylon a triangle with sides measuring 3, 4, and 5 units was used to measure right angles. In Egypt, right triangles were also used in their architectural constructions.

However, it was in classical Greece that the theorem was formally proved. It is attributed to Pythagoras of Samos and his school of mathematics for having found a rigorous mathematical proof of this theorem. The test was first performed by Pythagoras’s disciple Hippasus of Metapontus, although Pythagoras is said to have been the first to recognize its importance and practical applications.

The Pythagorean theorem became an essential tool for Greek geometry and It was used in the construction of temples, monuments and other architectural structures. The Pythagoreans also discovered many other interesting mathematical properties related to triangles, such as the relationships between the sides and angles of a triangle.

Throughout history, this theorem It has been used in various practical applications., from building construction to navigation and astronomy. It has also been a source of inspiration for mathematicians and philosophers, who have explored its symbolic meaning and its relationship with nature.

## What are the characteristics of the Pythagorean theorem?

The Pythagorean theorem has an explanation that can be summarized in certain quite specific characteristics, among some of the most outstanding we can find:

• Applies only to right trianglesthat is, those that have a right angle (90 degrees).

• the hypotenuse is always the longest side of the right triangle and is opposite the right angle.

• The theorem states that the sum of the squares of the two legs (the other two sides of the right triangle) is equal to the square of the hypotenuse.

• The theorem can be used to solve problems involving the length of an unknown side of a right triangleas well as to find the height of an object or the distance between two points.

• The Pythagorean theorem It has applications in physics. engineering and architecture, among other areas.

• It is commonly attributed to the Greek mathematician Pythagoras of Samos.who enunciated and demonstrated it in the 5th century BC

### What is the formula of the Pythagorean theorem?

The formula of the Pythagorean theorem establishes the mathematical relationship between the three sides of a right triangle. This formula is:

a² + b² = c²

Where ‘a’ and ‘b’ are the legs of the right triangle (the two sides that form the right angle) and ‘c’ is the hypotenuse (the segment opposite the right angle, which is the longest side of the right triangle).

This formula can be used to find the length of any unknown side of a right triangle, as long as the length of the other two sides is known. It can also be used for check if a triangle is right, since, if the formula of the Pythagorean theorem is fulfilled for the three sides of the triangle, then it can be affirmed that it is a rectangle.

It is important to remember that the Pythagorean Theorem formula is only applicable to right triangles, and cannot be used on other types of triangles.

## Examples of the use of the Pythagorean theorem in mathematics

This theorem has many very useful uses within the world of physics, geometry and mathematics and even economics. It should be noted that these uses can only be applied to right triangles. Some of the most important uses of this theorem are:

• Find the length of the hypotenuse: Suppose we have a right triangle with legs of length 3 and 4 units. To find the length of the hypotenuse, we can use the formula of the Pythagorean theorem:

c² = a² + b² c² = 3² + 4² c² = 9 + 16 c² = 25 c = √25 c = 5

Therefore, the length of the hypotenuse is 5 units.

• Find the length of a leg: Suppose we have a geometric figure, a right triangle with a leg of length 5 units and a hypotenuse of length 13 units. To find the length of the other leg, we can use the Pythagorean theorem formula:

a² = c² – b² a² = 13² – 5² a² = 169 – 25 a² = 144 a = √144 a = 12

Therefore, the length of the other leg is 12 units.

• Check if a triangle is right: Suppose we have a triangle with sides of length 3, 4, and 5 units. To verify if it is a right triangle, we can use the formula of the Pythagorean theorem to check if the relationship is true:

a² + b² = c² 3² + 4² = 5² 9 + 16 = 25 25 = 25

Since the relationship is fulfilled, we can affirm that the triangle is right-angled.