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El asombroso Teorema de Pitágoras

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Imagine that you want to know the height of this wall, but of course, it is very difficult to climb up and measure it with a ruler! How would you do it? Hmm… You can support a ladder, the length of which

You do know. You can also measure the distance between the support point of the ladder and the base of the wall. Is that enough? Well, a great Greek philosopher from more than 2,500 years ago discovered that, if you know these two sides, you can calculate the third with total precision

And without having to climb! Join us to understand… The amazing Pythagorean Theorem Pythagoras of Samos is known as the «first pure mathematician» and was born around the year 470 before our era. He formed a mathematical-magical-musical society that would be known as «the Pythagoreans» (although in his time they called themselves simply

«the mathematicians»). They saw that everything in the universe could be described by means of geometric and numerical relationships and proportions and therefore they gave mathematics a mystical and sacred character. For them numbers and figures, eternal and immutable, were the essence of earthly things, variable and perishable.

The Pythagorean school also developed a mathematical theory of music based on the relationship between the lengths of the strings and the notes they produce. They believed that each star emitted a note and all together formed the «harmony of the spheres» or heavenly music. Much of what the Pythagoreans studied

Was learned from Babylonian mathematics, including the famous Pythagorean Theorem, which apparently was known to the Chinese as well as the Babylonians. Let’s start by understanding the basics: the Pythagorean theorem applies to right triangles, that is, all triangles that have a right angle, that is, 90

Degrees. It doesn’t matter how long its sides are or what orientation it is in, if one of its angles is right, it is a right triangle. One of the sides, the one opposite the right angle, is called the hypotenuse. It is the longest side. By the way: the Greek word “hypotenuse” means “strongly

Tensed”: geometers used taut strings to perform their calculations. The other two sides, those that form the right angle, are called legs (leg means «that falls», because plumb lines were used to make measurements and ensure that a column, for example, was perpendicular to the ground). Ready? Well, the Pythagorean theorem

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Is a relationship between those segments and what it declares is that, in any right triangle, «the sum of the squares of the legs is equal to the square of the hypotenuse.» What does this mean? Easy: if you draw two squares: one whose sides are equal

To one of the legs; another whose sides are equal to the other leg and you add their areas, the result will be equal to the area of ​​a square whose sides are equal to the hypotenuse.

Let’s look at this example: on the right hand side, on the hypotenuse. that measures 5 units, we have a 5 x 5 square: we can see that it is made up of 25 1×1 squares. On this side,

One of the legs measures 4 units, that is, its square is 4 x 4 units: it is made up of 16 squares; the length of the other leg is 3 units, its square is 3×3, that is, it is made up of 9 squares. If we add these last two, 16+9, in total

They are… ah look! also 25 squares! (PAUSE OF 2 SECONDS) If we name each side of the triangle with a letter (conventionally put «c» to the hypotenuse and «a» and «b» to each of the legs), we can express this relationship as: a2 +b2=c2 (a squared plus b squared equals c squared).

This relationship is constant, which means that it “always happens”, regardless of the size of each of the sides of the triangle. For example: this other right triangle has its hypotenuse of 10 units and legs of 8 and 6 respectively. Let’s see if it is fulfilled?

8 squared (or 8×8) is 64. 6 squared (or 6×6) is 36. Added together they make 100, which is the same as 10 squared. Of course, in these examples we chose whole numbers, but this is not always the case: sometimes decimals come out. When the three sides are

Whole numbers, they are called a Pythagorean triple, like our examples: 3, 4 and 5 form a Pythagorean triple, as well as 6, 8 and 10 form another. But in this other triangle, one of the legs measures 7.5 units, and the other 4, which means that their squares are

56.25 and 16 respectively. Added together they give us the square of the hypotenuse: 72.25. Its square root is 8.5, which is the length of the hypotenuse. Since the three numbers are not integers , they do not form a Pythagorean triple. Let’s see in a very simple example how to use

The Pythagorean theorem algebraically to know the length of one of the sides when we only know the other two. Spiderman wants to save this baby, but he has little web left. He barely stops 25 meters! will he have enough to reach it? In this triangle

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We know that leg «a» measures 7 meters and leg «b» 24 meters. How long is the hypotenuse? If a2+b2=c2 then we know that 72 plus 242 equals c2. 7 squared is 49 and 24 squared is 576. Adding them together gives 625. 625 equals c squared. To

Know only the «c», without the square, we have to take the square root of the two terms: the square root of x squared, since it is x. And the square root of 625? Well, let’s find a number that multiplied by itself of 625… 25×25 gives 625. The result

Is 25! That is what the hypotenuse measures, that is, what the spider web should measure. Rescued baby! Let’s try our ladder and wall example. We know that the ladder measures 13 meters and that its support point is 5 meters

From the wall. That is, we know the measure of the hypotenuse and of one of the legs and we need to know the measure of the other leg. If we substitute them in the equation it would be a2+52=132. Since we want

To know the size of this leg, let’s leave that “a square”, which is our unknown, alone. I’m sorry, unknown, it’s for your own good! We do it by sending the 5 squared, which is adding, on the other side of the equation, but subtracting. It remains a2=132-52. 13 squared

Is 169 and 5 squared is 25. We subtract 169 minus 25 and we get 144. a2=144. The square root of “a” squared is “a” and that of 144 is 12. And that’s it! The height of the wall is 12 meters. Easy! See, cat? Now we know you better.

We leave you with a challenge: Imagine that you are going to build a house. Can you calculate how long this sloped roof should be, knowing that one of the poles is 2 meters, the other 3 and the base 4? In this case you have to do some previous operations. Pause the video

And leave us the answer in the comments. Something interesting about this relationship described by Pythagoras is that it is a theorem: that is, a proposition that can be demonstrated, and this theorem has been fully proven: it works with all right triangles

On flat surfaces: it is as absolute a truth as can be to have. But don’t think that mathematicians try to see if it applies to each of the infinitely many possible triangles, one by one: they rather design proofs that prove its general truth. It is said that

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In the universities of the Middle Ages, if you wanted to achieve the degree of Magister Mathesos you had to invent your own proof of the Pythagorean Theorem. Throughout history many people, including Persians, Arabs and India, have designed

Very ingenious proofs and the Pythagorean theorem is one of the theorems with the most different proofs. There is an ancient Chinese demonstration, another by Anairizi of Arabia, another by Euclid, one by Leonardo Da Vinci and one that Einstein made at the age of 12. Even the President of the United States

James Abram Garfield developed a proof and even high school students have contributed new proofs! What is the use of knowing the Pythagorean Theorem? Oh well, for many things! As you can imagine, it is used in construction and architecture to divide land or resolve any diagonal. In civil engineering,

To calculate the height of mountains and the inclination of roads. Also in naval and air navigation. Did you already know that the meshes of the 3D models used by video games and animated characters in movies are made of triangles?

This is because it is the simplest geometric figure and it requires less data to know everything about it. Because, if you know that the interior angles of a triangle always add up to 180 degrees, you can use trigonometric functions to find out everything about a right triangle,

Starting only from the measure of one leg and one of its other angles. A curious and very useful fact: although there are many triangles that are not right, if in any of them you draw from one of its vertices a line perpendicular

To the opposite side of that angle (PAUSE) you will always get two right triangles! Curiously Pythagorean! Another curious fact is that Platzi already has a TOEFL preparation course! Now, in addition to learning English at Platzi, you can also prepare to take the exam, manage your time and improve your

TOEFL performance, and you will also get a 15% discount on your exam. The TOEFL is one of the most important exams and a requirement to apply for some jobs, enter foreign universities, apply for scholarships and migrate to other countries. Show your

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