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  • What do Euclid,

  • twelve-year-old Einstein,

  • and American President James Garfield have in common?

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

  • the rule that says for a right triangle,

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

  • is equal to the square of the hypotenuse.

  • In other words, a²+b²=c².

  • 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.

  • The theorem is named for Pythagoras,

  • 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

  • that satisfy the theorem.

  • 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,

  • that has to make a right triangle,

  • and, therefore, a square corner.

  • And the earliest known Indian mathematical texts

  • written between 800 and 600 B.C.

  • 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?

  • Because we can prove it.

  • 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

  • and hypotenuse length c.

  • Arrange them so that their hypotenuses form a tilted square.

  • The area of that square is c².

  • Now rearrange the triangles into two rectangles,

  • leaving smaller squares on either side.

  • The areas of those squares areand b².

  • Here's the key.

  • The total area of the figure didn't change,

  • and the areas of the triangles didn't change.

  • So the empty space in one, c²

  • must be equal to the empty space in the other,

  • a² + b².

  • Another proof comes from a fellow Greek mathematician Euclid

  • and was also stumbled upon almost 2,000 years later

  • by twelve-year-old Einstein.

  • 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.

  • So for these three similar triangles,

  • you can write these expressions for their sides.

  • Next, rearrange the terms.

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

  • ab²+ac²=bc²,

  • or a²+b²=c².

  • Here's one that uses tessellation,

  • a repeating geometric pattern for a more visual proof.

  • Can you see how it works?

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

  • Here's the answer.

  • The dark gray square is

  • and the light gray one is b².

  • The one outlined in blue is c².

  • Each blue outlined square contains the pieces of exactly one dark

  • and one light gray square,

  • proving the Pythagorean theorem again.

  • 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,

  • ranging from brilliant to obscure.

  • Can you add your own to the mix?

  • Did you enjoy this lesson?

  • If so, please consider supporting our non-profit mission by visiting PATREON.COM/TEDED

What do Euclid,

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【TED-Ed】How many ways are there to prove the Pythagorean theorem? - Betty Fei

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    Anita Lin posted on 2017/09/12
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