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Teorema de Pitágoras - Contenido educativo

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Subido el 18 de julio de 2023 por Marta C.

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Demostraciones del Teorema de Pitágoras en un vídeo de TED Ed subtitulado en español.

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What do Euclid, 12-year-old Einstein, and American President James Garfield have in 00:00:00
common? 00:00:14
They all came up with elegant proofs for the famous Pythagorean theorem, the rule that 00:00:15
says for a right triangle, the square of one side plus the square of the other side is 00:00:20
equal to the square of the hypotenuse. 00:00:26
In other words, a-squared plus b-squared equals c-squared. 00:00:29
This statement is one of the most fundamental rules of geometry and the basis for practical 00:00:33
applications like constructing stable buildings and triangulating GPS coordinates. 00:00:38
The theorem is named for Pythagoras, a Greek philosopher and mathematician in the 6th century 00:00:45
BC, but it was known more than a thousand years earlier. 00:00:51
A Babylonian tablet from around 1800 BC lists 15 sets of numbers that satisfy the theorem. 00:00:55
Some historians speculate that ancient Egyptian surveyors used one such set of numbers, 3, 00:01:03
4, 5, to make square corners. 00:01:09
The theory is that surveyors could stretch a knotted rope with 12 equal segments to form 00:01:12
a triangle with sides of length 3, 4, and 5. 00:01:17
According to the converse of the Pythagorean theorem, that has to make a right triangle, 00:01:22
and therefore a square corner. 00:01:27
And the earliest known Indian mathematical texts, written between 800 and 600 BC, state 00:01:29
that a rope stretched across the diagonal of a square produces a square twice as large 00:01:36
as the original one. 00:01:42
That relationship can be derived from the Pythagorean theorem. 00:01:43
But how do we know that the theorem is true for every right triangle on a flat surface, 00:01:49
not just the ones these mathematicians and surveyors knew about? 00:01:54
Because we can prove it. 00:01:57
Proofs use existing mathematical rules and logic to demonstrate that a theorem must hold 00:01:59
true all the time. 00:02:04
One classic proof, often attributed to Pythagoras himself, uses a strategy called proof by rearrangement. 00:02:06
Take four identical right triangles, with side lengths a and b, and hypotenuse length 00:02:13
Arrange them so that their hypotenuses form a tilted square. 00:02:21
The area of that square is c squared. 00:02:25
Now rearrange the triangles into two rectangles, leaving smaller squares on either side. 00:02:28
The areas of those squares are a squared and b squared. 00:02:35
Here's the key. 00:02:39
The total area of the figure didn't change, and the areas of the triangles didn't change. 00:02:40
So the empty space in one, c squared, must be equal to the empty space in the other, 00:02:47
a squared plus b squared. 00:02:53
Another proof comes from a fellow Greek mathematician, Euclid, and was also stumbled upon almost 00:02:57
2,000 years later by 12-year-old Einstein. 00:03:02
This proof divides one right triangle into two others, and uses the principle that if 00:03:06
the corresponding angles of two triangles are the same, the ratio of their sides is 00:03:11
the same, too. 00:03:16
So for these three similar triangles, you can write these expressions for their sides. 00:03:18
Next, rearrange the terms. 00:03:27
And finally, add the two equations together and simplify to get a b squared plus a c squared 00:03:38
equals b c squared, or a squared plus b squared equals c squared. 00:03:46
Here's one that uses tessellation, a repeating geometric pattern, for a more visual proof. 00:03:57
Can you see how it works? 00:04:02
Pause the video if you'd like some time to think about it. 00:04:04
Here's the answer. 00:04:09
The dark gray square is a squared, and the light gray one is b squared. 00:04:10
The one outlined in blue is c squared. 00:04:15
Each blue outlined square contains the pieces of exactly one dark and one light gray square, 00:04:18
proving the Pythagorean theorem again. 00:04:25
And if you'd really like to convince yourself, you could build a turntable with three square 00:04:28
boxes of equal depth connected to each other around a right triangle. 00:04:32
If you fill the largest square with water and spin the turntable, the water from the 00:04:36
large square will perfectly fill the two smaller ones. 00:04:40
The Pythagorean theorem has more than 350 proofs and counting, ranging from brilliant 00:04:45
to obscure. 00:04:51
Can you add your own to the mix? 00:04:52
Idioma/s:
en
Idioma/s subtítulos:
es
Autor/es:
TED Ed
Subido por:
Marta C.
Licencia:
Reconocimiento - No comercial - Compartir igual
Visualizaciones:
6
Fecha:
18 de julio de 2023 - 17:27
Visibilidad:
Clave
Centro:
IES CARMEN CONDE
Duración:
04′ 56″
Relación de aspecto:
1.78:1
Resolución:
1280x720 píxeles
Tamaño:
20.94 MBytes

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