One of the most used and beautiful theorems in math is the Pythagorean theorem. Named after the greek mathematician who often is credited the theorems first proof it is one of the most well known mathematical theorems in the world. This site is dedicated to this theorem and to explain visually and theoretically how it works, how to use it and how to proof it.

Contents

1 Understanding the Pythagorean theorem
1.1 The theorem
1.2 A brief history
2 Basic examples where Pythagoras theorem is used
2.1 Find the length of the hypotenuse in a right triangle
2.2 Find the length of a leg in a right triangle
2.3 Is it a right triangle?
2.4 What is the diagonal distance across a square?

1. Understanding the Pythagorean theorem

The Pythagorean theorem or Pythagoras’ theorem is a relationship between the sides in a right triangle. A right triangle is a triangle where one of the three angles is an 90-degree angle. In a right triangle the sides are called legs and hypotenuse.

1.1 The theorem

the pythagorean theorem

In a right triangle the relationship between the legs a och b and the hypotenuse is

$a^2+b^2=c^2$

We can calculate c by using

$c=\sqrt{a^2+b^2}$

1.2 A brief history

Sculpture of Pythagoras

The theorem is often credited to the greek mathematician Pythagoras (c. 570–495 BC) even if there aren´t any proof that he was the first one that proved the theorem.  The pythagorean theorem where also known by Babylonian and Chinese mathematicians before Pythagoras. It is not known whether the theorem were discovered once or many times in different places.

The history of the theorem is also connected to the discovery of pythagorean triples which consists of three positive integers such that $c^2=a^2+b^2$. The three integers 3, 4 and 5 is a well known pythagorean triple because $ 5^2=3^2+4^2 $. Pythagorean triples were discovered by the babylonians, from 2000 to 1786 BC, even though they never mentioned any triangles.

Pythagoras or the disciples to him constructed the first known algebraic proof of the theorem and famous writers such as Plutarch and Ciceron acclaimed him for discovering this proof. Therefore this beautiful connection between the sides in a right triangle has been credited to him.

Sources for this section: Wikipedia eng, Wikipedia swe 

2 Basic examples where Pythagoras theorem is used

2.1 Find the length of the hypotenuse in a right triangle

Find $c$

Solution:

$ c^2 = 3^2+4^2 $

Simplify the right side

$ c^2 = 9+16 $

$ c^2 = 25 $

Take the square root

$ c = \sqrt{25} = 5 $

The hypotenuse is $ c = 5 $

2.2 Find the length of a leg a right triangle

Find $a$

Solution:

$ 13^2 = a^2+5^2 $

$ 169= a^2+25 $

Substract 25 from both sides

$ 169-25 = a^2 + 25-25 $

$ 144 = a^2 $

Take the square root

$ a = \sqrt{144} = 12 $

The leg is $ a = 12 $

2.3 Is it a right triangle?

The sides in a triangle is 6, 12 and 13. Is it a right triangle?

Solution:

The side that has the length 13 has to be the hypotenuse if it´s a right triangle.

Let´s use the pythagorean theorem to check if $13^2=6^2+12^2$

$13^2=169$

$ 6^2+12^2=180 $

This cannot be a right triangle because  $13^2≠6^2+12^2$

(if the first leg is 5 instead it would have been a right triangle)

2.4 What is the diagonal distance across a square?

The sides in a square is $1$. Which is the diagonal distance in the square?

Using pythagoream theorem to calculate the diagonal distance in square

Solution:

The diagonal distance ca be calculated by using the pythagorean theorem since it is the hypotenuse in a right triangle with the legs $1$ and $1$.

Lets call the hypotenuse $c$ which gives us that

$c^2=1^2+1^2$

$c^2=2$

Take the square root

$c=\sqrt{2} ≈ 1,414 $

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