Name: 【TED-Ed】How many ways are there to prove the Pythagorean theorem? - Betty Fei
Uploaded: 2017-09-12T13:45:17.000Z
Duration: 5 min 17 s
Description: Thousands of YouTube videos with English-Chinese subtitles! Now you can learn to understand native speakers, expand your vocabulary, and improve your pronunciation...

and American President James Garfield have in common?

They all came up with elegant proofs for the famous Pythagorean theorem,

the square of one side plus the square of the other side

is equal to the square of the hypotenuse.

This statement is one of the most fundamental rules of geometry,

and the basis for practical applications,

like constructing stable buildings and triangulating GPS coordinates.

a Greek philosopher and mathematician in the 6th century B.C.,

but it was known more than a thousand years earlier.

A Babylonian tablet from around 1800 B.C. lists 15 sets of numbers

Some historians speculate that Ancient Egyptian surveyors

used one such set of numbers, 3, 4, 5, to make square corners.

The theory is that surveyors could stretch a knotted rope with twelve equal segments

to form a triangle with sides of length 3, 4 and 5.

According to the converse of the Pythagorean theorem,

And the earliest known Indian mathematical texts

state that a rope stretched across the diagonal of a square

produces a square twice as large as the original one.

That relationship can be derived from the Pythagorean theorem.

But how do we know  that the theorem is true

for every right triangle on a flat surface,

not just the ones these mathematicians and surveyors knew about?

Proofs use existing mathematical rules and logic

to demonstrate that a theorem must hold true all the time.

One classic proof often attributed to Pythagoras himself

uses a strategy called proof by rearrangement.

Take four identical right triangles with side lengths a and b

Arrange them so that their hypotenuses form a tilted square.

Now rearrange the triangles into two rectangles,

The areas of those squares are a² and b².

The total area of  the figure didn't change,

and the areas of the triangles didn't change.

must be equal to  the empty space in the other,

Another proof comes from a fellow Greek mathematician Euclid

and was also stumbled upon almost 2,000 years later

This proof divides one right triangle into two others

and uses the principle that if the corresponding angles of two triangles are the same,

the ratio of their sides is the same, too.

you can write these expressions for their sides.

And finally, add the two equations together and simplify to get

a repeating geometric pattern for a more visual proof.

Pause the video if you'd like some time to think about it.

Each blue outlined square contains the pieces of exactly one dark

And if you'd really like to convince yourself,

you could build a turntable with three square boxes of equal depth

connected to each other around a right triangle.

If you fill the largest square with water and spin the turntable,

the water from the large square will perfectly fill the two smaller ones.

The Pythagorean theorem has more than 350 proofs, and counting,

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