What is the Pythagoras’s Theorem for?

short answers for big questions

The Pythagoras’s Theorem is one of the most known mathematical theorems. It comes after the Greek mathematician Pythagoras of Samos who was not its inventor or discoverer (because the theorem was already known a long time before the appearance of Pythagoras) but he was the first mathematician to show its truthfulness. In order to easily memorize this theorem here it is a short rigmarole my teachers taught me when I was a pupil at school and I have never forgotten about it:

"Pythagoras of Syracuse
told his grandchildren some day:
The square of the hypotenuse
is equal to the sum of the cathetus."

But why is this theorem so important?

Because of a simple reason: if you know two measures of the sides of a rectangle triangle, it becomes easier to find the measure of the side that misses. The importance of this theorem is more evident when we notice that we are surrounded by rectangle triangles everywhere, since a good deal of nature (a tree draws a right angle with the soil) as well as human constructions (buildings, bridges, monuments) are based on right angles.

How do you recognise the cathetus and the hypotenuse?

It´s easy. The biggest side corresponds to the hypotenuse and it is the one opposite to the right angle. The other two sides that make up the right angle are the cathetus.

nomenclature of the right triangle

So, what is the famous formula of the Pythagoras’s theorem?

Having in mind what has been said in the above rigmarole, the mathematical formula is the following: `H^2 = C_1^2 + C_2^2`

How do you prove the formula is valid for every rectangle triangle?

Out of curiosity it is interesting to know that the book "The Pythagorean Proposition" is composed of 370 different demonstrations concerning the Pythagoras’s theorem. So as not to be too technical, instead of difficult demonstrations to be understood, I will leave here two small animations which are good enough to illustrate that the Pythagoras’s theorem formula is valid for every rectangle triangle.

Demonstration of the veracity of the Pythagorean theorem
Demonstration of the veracity of the Pythagorean theorem

I have already heard about the Pythagorean ternary: what´s that about?

It is about a set of three positive and whole numbers, that is, natural numbers (they are sometimes known as Pythagorean trio or Pythagorean triple) from which it is possible to “build” a rectangle triangle with those measures. So as to give just an example here it is the first five Pythagorean ternaries: `(3,4,5)`; `(5,12,13)`; `(7,24,25)`; `(8,15,17)` e `(9,40,41)`.

Can I use the Pythagoras’s theorem in any triangle?

No, you can’t. The Pythagoras’s theorem can only be used in rectangle triangles, that is, triangles which are formed by an angle of 90º. Fortunately, it is possible, in almost every polygon, to divide them so as to obtain one or several rectangle triangles. For instance, a square is composed of two equal rectangle triangles.

But what happens if a triangle is not rectangle?

Well, in this case you have to use another law known as the cosine law, which enables us to calculate the third side of every triangle, since we know the length of two of its sides and the angle formed by them. The Pythagoras’s theorem is a particular case of this law!

escrever carta

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