Pythagorean Theorem Proof (Geometry)
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- Опубликовано: 19 сен 2014
- The Pythagorean Theorem says that for any right triangle, a^2+b^2=c^2. In this video we prove that this is true. There are many different proofs, but we chose one that gives a delightful visual explanation for why the Pythagorean Theorem works.
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Geometer: Louise McCartney
Artwork: Kelly Vivanco
Written by Michael Harrison
Directed & Produced by Kimberly Hatch Harrison & Michael Harrison
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![Pythagoras Would Be Proud: High School Students' New Proof of the Pythagorean Theorem [TRIGONOMETRY]](http://i.ytimg.com/vi/p6j2nZKwf20/mqdefault.jpg)
![Pythagoras Would Be Proud: High School Students' New Proof of the Pythagorean Theorem [TRIGONOMETRY]](/img/tr.png)
People should be taught this in school instead of just being given the formula and told to memorize it.
I agree... it's truth
damn right... i feel like maybe the teachers themselves didn't even know, and were told to memorize it when they were in school... :}
in which they probably had to re-look over for when they became teachers... lol
i could do so much for the high schoolers if i were a teacher... understanding sine, cosine and tangent is one of these things you don't get explained either...
you just get told to memorize sohcahtoa, so that you know what inputs to press on the calculator to figure out angles and sides... but do they bollocks have a clue why...
i only learned once i became a programmer...
even when i first learned to use them with the rotation algorithm, i didn't understand the logic behind why the rotation works, but i do now...
it also leads onto dot products of vectors... which can be used to project the length of a vector perpendicular to another... :]
anyway, they probably feel they'd be wasting their time, since yeah, do they all need to know all this stuff? what if they end up working in gregs? lol
***** yeah, i know... :]
but algebra wouldn't even seem like anything scary and hard to those wimpy girl kinda students, if say, you were viewing it in the way of programming...
banana = 8
therefore, we can split that banana...
split = banana / 2
xD
jk, but it may possibly help... at least make IT fucking useful in school... we all know how to use computers... xD
if your mind was curious enough you would have proposed to find the reason and truth yourself rather than blaming your teachers...i don't blame you
After finishing my Engineering degree, where I use the right angle triangle a million times, I finally found out WHY we can do this haha
Yeah I figured out a proof of the quadratic formula as well only after I got my degree. That one used to have me perplexed.
@@JSSTyger It is sad! Our education system failing us
@@EliA-mm7ul I bottomed out of math and science because I cannot remember what I dont understand. Also the mathematical mode of expression is just gibberish to me. I was told often just remember the equation and forget about why. However, I also realized that if I had to understand every equation I used, I would spend my whole life learning and not getting anything done. The 3000 year history of math is too big to be taught meaningfully. Using formulae you dont grasp is just essential for getting the job done. Even tho its not the job for me.
3:24 if you confused by what happened here, (a+b) squared is basically (a+b)(a+b) and if you distribute that, a times a is a squared, + a times b which is ab, + b times b which is b squared, plus b times a which is ba, or ab. So now we have the terms a squared. b squared. ab. and ab. Since were trying to add them all together ab plus ab is just 2 times ab. 2ab. So now its 2ab + b squared + a squared. Or in her case using associative property of addition she moved around the terms. That's how she got that equation.
Thank you! This was so helpful!
Beautiful presentation of this elegant proof. Thank you!
They never showed you these proofs in school but i loved to go look them up and i loved being able to understand for the first time
Aw I love this proof. So simple but the fact you can visualise it is nice
This class, it's so completed. I've learned more about it that has showed the foundation of this theorem. Thanks ...
GOOD EXPLANATION WITH BEAUTIFUL DIAGRAMS ,SPEECH AND EXPRESSIONS
...
Beautifully done! The shortest most elegant and precise proof of the Pythagorean Theorem. I am almost sad that summerhollidays are coming... :D :D
Right-on! Unfortunately I find myself at the end of another Socratica playlist; I’ll have to find another to binge-watch!
It's just as good as Euclid's proof that there are infinitely many primes. Proofs are the best thing in mathematics.
Thank you!!! Keep these videos going!!!
I've never seen it expressed like this, nice work.
awesome explanation guys .Keep up the good work
Super High Five & Quod Erat Demonstrandum Gorgeous !
Well done Socratica, Please keep up the awesome work.
Best Regards,
Kenneth.
Thanks so much, Ken Lee !!
Really one of the best videos on the topic.
Elegant! - with the least assumptions and using the very basics of algebra and Geometry.
Visual in Combination with theory is Awesome.
Your aesthetics equal your timbre and your articulate way you explain this proof.
Good video, best I have seen. Here is an important tip. The Pythagorean Theorem works in 3 dimensional space. So, consider x,y, and z (elevation coordinates). Then x squared + y squared + z squared = length squared... So for example. imagine a squared off room. What is the length from a corner to a spot on the ceiling somewhere in the room? I you know its x,y and z coordinates...use the Pythagorean theorem. I use this in machining, outside measurements, etc.
thank you soooo much,I was explaining it to my kid and it was so helpful thanks again
Great video, beautiful explanation. Thanks
Nice video. It really helped me a lot. Thank u so much .
Another awesome video, Keep up the great work.
I love her explanation. It's very under understandable.
Ur so calm, I love Ur videos.
😻❤ thank u
Oh my gosh this helped so much! I had math homework that asked me to prove Pythagoras theorem and it made no sense to me at all. I watched other videos but they didn't really help at all! I used this and it totally made it so much easier for me to understand! Thank you!
+Lucy Hendry That's wonderful! We're so glad to hear our video helped. Good luck with your math class, and come see us again! :)
there are at least 300 hundred differtent ways to demonstrate Pythagore Theorem
Well done. Good Explanation
It's nice to see a method that uses algebra and not just rearranging shapes.
thanks...that is was very helpful
Could you do a proof for De Gua's theorem?
Thanks it's very helpful
thank you it helped me a lot
Was really helpful.
What is a good mathematical journal to present a new proof of the Pythagorean Theorem? I proof it differently and have checked with more than 400 solutions and did not find my solution, so I want to submit it for publication. Can you suggest a journal?
Reader's Digest
As The Truthful Channel noted, this proof should be taught in all high school geometry classes. First, it's not difficult to understand once the trick is laid out for you as in this video. Second, a simple proof like this demystifies the theorem and gives the students the feeling they're not expected to be passive robots tasked with memorizing formulas of dubious utility. Third, the proof gives insight into mathematical ingenuity. An instructor can instill the feeling that every single student in the class could have come up with this proof with just a little perseverance and willingness to think. You only need to know a bit of algebra and the area formulas for squares and triangles. Such an approach, which avoids the subtle implication that students are intellectually incapable or too disinterested to understand the origin of famous results, can provide an early trigger to a mathematically talented students who otherwise might never have considered studying the subject seriously.
And this is exactly one of the reasons I lost interest in math at a young age when I was suited for it...
Thank you …. Very well done !!
THANKS IT HELPED ME VERY MUCH
Thank you. very simple.
BRAVÍSIMOOOOOOOOOOOOOOOOOOOOOO!!
AWESOME!!
Thanks ever so much.
Excellent video lecture
Nice explanation
Thank you this helped so much!
You are so welcome! Thanks for letting us know! :)
Thanks for helping me to learn it.
We're so glad we could help! Thanks for letting us know - that really motivates us to keep making videos. 💜🦉
This is a much better way of proving Pythagoras,s theorem rather than just saying that a squared plus b squared equals c squared which is not in don't as can easily be shown but that in it self don't prove that the triangle is a right angled but your method makes this clear .
THIS ACTUALLY MADE SENSE THIS IS AWESOME
+Chloe Liu That's so great to hear! Thanks for watching! :)
Good explanation
We had to learn this for our exams, I spent about 2 hours last night trying to remember it and all I got was a bunch of ultimately useless truths about a right triangle by using sin cos and tan of two triangles made using H as an adjacent line to the hypotenuse through the 90deg angle of the original right triangle. Gave up once I got an equation that boiled down to 0 = 0
Now i understand.i now have an assignment.
Thank you so much.
This is awesome...
Is there a name for this proof? Also, it would be nice if you could explain why the proof fails when the triangle is not a right triangle. Presumably because you cannot form a square when you align them?
I don't know of a name for this proof. (There are actually quite a few proofs of the Pythagorean theorem!) You're right, for non-right triangles, this proof doesn't work - you can't make a square. You can, however, generalize the Pythagorean Theorem to work with *any* triangle. This gives you the law of cosines from trigonometry. We'll make a video about this eventually. :)
Pythogoras theorem is valid for right triangle only
Software Architecture & Design this is called Bhaskara's proof
Software Architecture & Design
Beautiful.
Nice, thank you.
Pls ma a full video on Pythagoras rule from the beginning to the end pls a video ma pls reply I need it aloy
If this had been my Maths teacher, I would have been the best student in my class.
Please answer me if you can:
Do we know why the circle has 360 degrees and why is 1 degree the foundation of common angle measurements?
Because in all these gemotric theorums etc. we use these as the base, unless the ancients proved them in some other fashion.
360 = 2*2*2*3*3*5. So it is easily divisible by 2 and 10 and other useful numbers. It always struck me that there are 365 days in a year and 360 degrees in a circle. Close enough for Flint Stone astronomy. In engineering we generally use radians for angle. There are always 2 pi radiuses around a circle (360 degrees) and pi radiuses in 180 degrees and pi/2 radiuses in 90 degrees. A weird number but not at all arbitrary. If the radius is 1, you can just say 2 pi, pi and pi/2 and that is the actual length you travel around the arc associated with that angle. Oh, we actually call them "radians" not "radiuses".
360 deg is an arbitrary choice, although not so arbitrary if you think that the Babilonians thought the year had 360 days. I think that the military use 400 deg circle (?)
What a nice teacher
Brilliant!
Great video but how do you multiply out the (a+b)² to get a²+2ab+b² ?
+DarkDuke 2015 Thanks for watching! Have you done this kind of multiplication before? Try writing it out like this: (a+b) x (a+b).
Then be really methodical and careful, and multiply out each part in order: axa = a^2; axb = ab, bxa = ba (or ab, same thing), and bxb = b^2. add it all up and you get a^2 + 2ab + b^2.
(do you see how when you add ab and ab you get 2ab).
Be careful with this kind of multiplication if there are any negative signs, because then some of the parts will be negative and you'll have to subtract instead of add. This one had all + signs, so you just had to add all the parts together.
Hope that helps!
Socratica Cool thanks i watched a video the other day about that he called it foil. I do get it. You can also redraw the rectangle of (a+b)² and draw in the lines. a² , b² , and the 2ab fit into it perfectly =)
Nice! Yes, "foil" is a good mnemonic for remembering to do all the parts of the multiplication (First, Outer, Inner, Last). Glad to hear it is all working for you! :)
@@Socratica Nice explanation
Did thay not teach you that?
Try this:
0:22 a²+b²=(a+b)²-2ab
Consider: c²+2ab
Factoring:
c²(1+2(a/c)(b/c))=c²(1+2sinAcosA)
The value 2ab/c² is the double angle
trig ratio sin2A.
Consider: (a+b)²/c²
This is equivalent: (sinA+cosA)²
Expanding out: sin²A+cos²A+2sinAcosA
Subs for 2sinAcosA & due to “†” & “‡” (bottom), this is: 1+sin2A.
From “1”, have: c²+2ab=c²(1+sin2A)
From “2”, have: (a+b)²/c²=1+sin2A.
Therefore:
c²=(c²+2ab)/(1+sin2A)=(c²+2ab)/(a+b)²/c²
Multiplying by (a+b)²/c² on both sides:
(a+b)²=c²+2ab => (a+b)²-2ab=a²+b²=c²
…QED 😊.
(†)From the compound angle formulas(‡)
for sin & cos, you get double angle formulas for those, and from cos2A you get:
1-2sin²A=2cos²A-1=cos2A
=> 2=2(cos²A+sin²A)
Therefore: 1=(cos²A+sin²A)
(‡)Proof in this video:
“Angle sum identities for sine and cosine” by blackpenredpen
2 new orleans high school students is using this same explanation to say they've proven the theorem.
I read that their explanation involves the Law of Sines, I did not see sines mentioned at all in this video but I only skipped through it in order to see if sines feature in it. Are you sure?
Le théorème de Pythagore s’obtient directement à partir d’une propriété du triangle rectangle selon laquelle on engendre deux triangles semblables au triangle d’origine en projetant une droite à partir d’un de ses sommets sur le côté opposé. On montre facilement que le triangle rectangle est le seul à avoir cette caractéristique. L’hypoténuse est alors divisée en deux par cette ligne projetée qui n'est autre que la hauteur partant de l'angle droit. Chaque angle non droit peut alors s’interpréter de deux façons, soit avec l’hypoténuse en entier soit avec une partie seulement, auquel cas c'est l’autre côté qui est cette fois-ci l’hypoténuse du triangle rectangle inscrit. Voilà d’où vient a^2, le côté a est tantôt hypoténuse tantôt non hypoténuse, donc tantôt au numérateur tantôt au dénominateur dans la formulation issue des triangles semblables, les produits en croix donnent a^2. On obtient le même type d'équation avec l’autre angle non droit, l’addition des deux égalités donne le théorème de Pythagore. Bien à vous
Very good
Nice... thanks
Oh my god, how do people think of so good constructions, I can't even think a Little.
We'll done!
That's a really intuitive proof. I always see people draw squares just on the sides of the triangle and summing their area together and the areas being the same. Thats just a circular argument though and it bothered me.
what proof is this?
Does anyone know the name of this proof?
Pythagoras theorem proof there is no name
thank you :)
Oh Its great...why dont the teachers teach like this instead of giving the formula directly...thank u so much..Raj from India
This woman is a stunner....
I've noticed that all of the people your channel feature are beautiful.
Btw, your channel is AWESOME!
+The Wireless Brain Aw, thanks so much for your nice comments, and thank you for watching! :)
Socratica
No problem! :D
this helps a lot as an Indian student
I'm going to fail
Lmao
did you?
Thank you :)
GothicDarkhellrazor
To be a bit more creative.... rather then just using areas...
a^2 + b^2 = c^2
rearrange
a^2 = c^2 - b^2 = (c + b)(c- b)
a*a = (c + b)(c- b)...... intersecting chords
a/(c- b) = (c + b)/a ....... similar triangles
Great!..
i love this.
Amazing
Thanks
never ever took interest in the proof........... thanks a lot very interesting .........
nice 1
En cuanto al Triángulo rectángulo, donde se aplica el "Teorema de Pitágoras", tengo un nuevo concepto sobre este "Teorema" voy a llamar "Teorema de Sidney Silva" sigue mi relato;
Condición de Existencia de un Triángulo; para construir un triángulo no podemos utilizar ninguna medida, tiene que seguir la condición de existencia: Para construir un triángulo es necesario que la medida de cualquiera de los lados sea menor que la suma de las medidas de los otros dos y mayor que el valor absoluto de la diferencia entre estas medidas, esto está relacionado en el Teorema de Pitágoras, ya en el "Teorema de Sidney Silva" podemos sí utilizar cualquier medida, siguiendo la condición de existencia; donde puedo construir un Triángulo que la necesidad de la medida de cualquiera de los lados sea mayor que la división de las medidas del lado más pequeño del valor absoluto (fórmula a ^ 2 = b ^ 2: c ^ 2 o b ^ 2: c ^ 2 En el caso de la hipotenusa será menor que los catetos, y siempre los catetos serán mayores que la Hipotenusa, donde los números 5,4,3 ya están obsoletos, cuando cambien de números, será aproximado , redondeado y simplificado, ya por mi "Teorema de Sidney Silva" siempre será exacto con 100% exactos .. !!!!, Sr Sidney Silva.
Eso es un corolario del Teorema de Pitagoras, no un Teorema. Buen intento
Wow m a student nd its helpfull for me
hi beb
Wow very simple.
perfect
thanks beyb
The Pythagorean Theorem is not only a famous theorem, but the hardest to pronounce.
I have a disability that makes difficult to remember things. So the way solve this problem is by understanding. And this an prime example.
awesome
Nice
My mind is blown 🤯🤯
1:02 Not random; arbitrary!
fortunately the most beautiful proof of Pythagorean theorem and the female speaker too.
There is another proof that is entirely geometrical. No algebra required. Just move the four triangles into a different configuration within the larger square.
nice
Peace😊 Love from India🙏
How is the small one a square
Faheem Ahmad They took 4 congruent right angled triangle so each C is equal