Pythagorean Theorem & Its Inverse (my favorite proof)
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- Опубликовано: 31 янв 2019
- I'll be sharing my favorite proof of the Pythagorean Theorem and its inverse, using similar triangles. The Pythagorean Theorem is a fundamental concept in geometry that states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. But why does this theorem hold true? In this video, I'll walk you through a simple and elegant proof that uses similar triangles to demonstrate the Pythagorean Theorem and its inverse. By understanding how the ratios of the sides of similar triangles relate to each other, we can prove this theorem and its inverse with ease. Whether you're a math enthusiast or just curious about the Pythagorean Theorem, this video is for you.
Pythagorean triple generator 👉 • finding ALL pythagorea...
This question was asked by Toby from my Chinese channel:
中文版 • 畢式定理有兩種: a^2+b^2=c^2 &...
@blackpenredpen
![Pythagoras Would Be Proud: High School Students' New Proof of the Pythagorean Theorem [TRIGONOMETRY]](http://i.ytimg.com/vi/p6j2nZKwf20/mqdefault.jpg)
Omg spiderman is supporting your channel!! You're going to be at 400k subscribers by the end of the year for sure
René Gado yup, it's very nice of him!!!
And someone special too
Sir please make videos related to all types of figure counting. Need for competitional exams 😑
It's stuck at 301K
Now it's 6.81lakh🥳
This is far clearer and more straight-forward than the traditional proofs.
Nathan Greene 😊
Possibly not as intuitively motivating, but the logic is as pristine as a crisp winter morning. There, who's not a poet?
Oh dude search president Garfield s proof
The inverse pythagorean theorem is some serious beautiful shit.
Brand new Maths for me. BTW, my 71st B. D. was this year 2019.
@@inyobill happy birthday, old one!
"Better old than dead", kiddo! And thanks for the kind wish. :-D
Ah lads don't worry about it, we all DIE!(Maniacal Laughter)
@@MrCoffeypaul What? What? Wait, what did he say?
The Pythagoras Lu! 😄
Pytha Lu? ;-)
Wise man once say triangle is strongest shape.
L H agree!!
Wow, I've never seen the inverse Pythagoras Theorem. Love your proof videos!
David Morris thanks!!!!
I love math so much! Thank you so much for this video. For Pythagoras, as soon as you drew the vertical height, h, I decided to try to do the proof and I did it. I also just happen to label the parts of c as c1 and c2 also lol. For the inverse, I knew what was going to happen as you were going and it was beautiful! The simplicity of math can be so great; I just wish schools showed it the right way.
FACH2004 hahaha very nice. Thanks
My favorite proof of the Pythagorean theorem is Garfield's (the president, not the cat) proof, just because of how clever it is.
DerivativeOfSenpi
Oh man!! I just searched it. That is amazing!!!
Yeah, I love that one too. Basically the same as the square + four triangles = bigger square one, just cut in half to get a trapezoid made of three right triangles. Write the area of the three triangles, write the area for the whole trapezoid, equate them, do some math, and you get the theorem!
@@blackpenredpen There's a video that I made in my channel that I would love for you to check out, about sec^2x. I promise you will like it. Thanks !
Yes, it's nice, but it's just another of those "draw a square and populate the corners with right-angled triangles and do some algebra" proofs. He just ignored half of the square.
On the other hand, it's always nice to know politicians haven't always been entirely useless (I speak as a Brit).
@@davidgould9431 James Abram Garfield, 20th U.S. President, and the 2nd President to be assassinated, had been an Ohio schoolteacher before entering politics.
So his non-political contribution to society went well beyond his PT proof.
Incidentally, as I read it, some decades ago, Garfield's proof *did* include the whole square, complete with 4 right triangles.
Which makes the algebra a bit more direct, as there are no ½'s to be canceled from both sides. There's only:
(a + b)² = 4·½ab + c²
a² + 2ab + b² = 2ab + c²
a² + b² = c²
Fred
Awh the nostalgia. This video hit me in the heart. It's amazing how far we've come from counting numbers to algebra and to functions and calculus and to complex. I remember not understanding the Pythagorean theorem and back then trigonometry was like the real deal.
I wish they taught this proof in schools to teach kids things don't just come out of the blue.
Loved the inverse proof, and also this is now my favourite proof as it uses similar triangles which is neat!
DavidAde yay!!!!
Indeed...totally agree considering that this proof is about a right triangle's feature and the fact it only relies on similar triangles (not squares or other polygons here) makes it my favourite demonstration of the PT as well. Beyond that I always keep it in mind as a special case of the cosine law which applies to all triangles.
Thank you!!! This was a beautiful proof!!! :-D
Love how you tackle really hard and also simpler problems with the same enthusiasm!
I like your videos because I can see that you're genuinely excited about the maths. There's a million people out there that understand it well enough to explain it, but it's a good video because you love it.
Cool! Thank you!
This proof doesn't ask for special math knowledge, just fundamental ones.
I have done it, but in different way:
I use sin and cos in my proof. After I finished, I have understood that sina^2 + cosa^2 = 1 is also Pythagorean theorem and I used Pythagorean theorem in my proof of Pythagorean theorem 👌
The Inverse Pythagorean theorem is really useful, I think
Hello!
This is clearly sorted out!
Thank you for sharing!
Keep up the good work!
Really pleased to see the inverse proof. And who, I wonder, are the four people who don't like this video?
I NEVER KNEW THE INVERSE ONE!!!!!
Very easy to understand. Love the teacher's enthusiasm!
Bravo! As always, super presentation!
This is a very interesting video. I subscribed!
Davi Nonnenmacher yay!! Thanks!
beautiful. the proof based on triangles similarity is really elegant.
Such a simple proof! I absolutely love it!
omg, I love that inverse to get the height from the hypot. Very nice, thank you!
Keith Masumoto yay!!!
Very simple and easy to teach demonstration of the famous theorem. Thanks a lot for providing brand smart different points of view!!!
this concept has already helped me in solving question thanks for covering it
Mind blowing proofs, i am speechless. Keep up the good work 👍👍👍
Nicholas Yap thanks!!
I really like this, as it gives a better explanation of proving the pythagorean theorem, and I love him showing the inverse. This will be a good way for me to help the students I work with, and will help with any geometry students I get working on similar triangles.
Enjoyed! Brilliant!
The most amazing proof of the Pythagoras I've ever seen. Thanks Steve!
Bro, I am so glad I subscribed to your channel. Cheers.
It's wonderful result!
I really like this proof as well! So much easy to visualize somehow.
Beautiful proof!
Well explained. I wasn't aware of this cool proof.
这些视频都太好了。 多谢！
Amazing explanation and video :)
That is an Awesome proof!
Last year on this day i subscribed this channel.... It has been an amazing journey with blackpenredpen #yay ...🖤
The inverse is too beautiful, I remember that elegance after deriving it myself :)
Very cool proof. I am a big fan.
Great! Simple, plain, non-obfuscated. Spot on!😎
Per Appelgren thank you!!
Excellent !
Good video!
That was AMAZING!
YORUMCU DRAVEN thanks!
I like the square proof you said you didn't like. I find getting the same thing two different ways is a cool method. Like, deriving the Sine and Cosine rules. Never knew about the inverse theorem and I'm a maths teacher myself. Very elegant and a great way to get the height quickly.
Superb!
Good job! 😇👍👍
This is the first proof I loved... Like we use it every day and it's fun
Thats a beatiful way to proof pythagorean thanks.
Beautifull proof and formula
Thank yyou for proove this theory to me at last
I love this proof!!
I remember have to find a proof for the Pythagorean theorem for a pre-calc project and I found on that used the area of a square and area of a triangle formulas
DAMN I LOVE THIS GUY HE IS AMAZING
Derivatives&integrations
Thank you!!
blackpenredpen YOU ARE AMAZING 😂 MATH IS MY FAVORITE LESSON AND U MADE ME TO LOVE IT EVEN MORE
@@snorlass oh same!!
That was so great i liked it.😘
What a simple proof,you Chinese, have turned the world upside down.you have unearthed everything.you are amazing.
Cool proof! I actually understood it lol.
Proof by area, really straightforward.
*Really really cool*
I love to using James Grifeld formula,and your are excellent too for proof for Pythagorean theorem
MAGNIFIQUE !!! MERCI INFINIMENT ...
They say the pythagorean theorem have 50 proof,but from now on ur first proof is my favorit.
There are more than 300 proofs...
I remember getting a 5 (best grade) in high school partially thanks to the inverse of the Pythagorean theorem (didn't know it was called that way though). In fact, using a bit of manipulation it can be further expanded to work for a regular three-sided pyramid: 1/H^2 = 1/a^2 + 1/b^2 + 1/c^2, where H is the height. This formula is super useful to find the height if you have the sides of the pyramid, otherwise you'll spend over an hour digging around.
What an awesome proof
This is cool!
Nice!!
The inverse is very useful.... lovely video
Interesting! I didn't really know there was an inverse PT and I pondered what it was going to be throughout the video because I was thinking in terms of inverse functions, but I often forget that inverse has the occasional meaning related to "reciprocal".
Beautiful Proof.
Mandeep Singh thanks
Nice one ! I like to the Full Square Area proof. :: Build a square with side s = a + b such as the right triangle is used 4 times. The middle area of that construction is a square of side c. Trivial proof : by construction. So, Surface S = s^2 = c^2 + 4T where T = ab/2 ( area of the triangle )Thus,S = s^2 = (a+b)^2 = c^2 + 4*(ab/2) a^2 + 2ab + b^2 = c^2 + 2ab a^2 + b^2 = c^2 [DONE]
I hadn't seen the inverse pythagorean theorem before, very nice! My favorite proof is just the square of side length a+b with the square of side length c touching it with the corners 'a' away from each outer corner : this gives you 4 abc triangles inside the square of size a+b, The whole square is area (a+b)^2, the inner square is area c^2 and there are 4 triangles area (a+b)/2 which add up area 2ab for the triangles.... So the whole outer square is a^2 + 2ab + b^2, but it it is also c^2 + 4(ab)/2 = c^2 +2ab, so a^2+2ab+b^2 = c^2 + 2ab, and you subtract 2ab from both sides.
Damn wow I've never seen the inverse formula before! That is definitely gonna be useful
xMinix cool!!!!
Nice proofs. You could actually get the relation ab=ch from the similar triangles too (from the first and the third triangle you have a/h=c/b).
this is the proof i got taught in school...always liked it more than the other ones
Very nice proof of Pythagorean theorem ! It seems to me I didn't see this proof elsewhere. Thank you !!
Jordan Jordan yay! You're welcome!
I believe Einstein came up with it when he was young
Jordan Saenz interesting!! I actually learned about this when I was in college. And this kinda similar triangles in a right triangle problem happen a lot on standardized tests such as the SAT
@@blackpenredpen Yeah I saw one or two problems about triangle ratios in the PSAT
This proof is part of the Indian 10th grade similarity syllabus :)
Wow you're very good thank u very much
This was majestic to watch 😢
Wow, very excellent reciprocal Pythagoras theorem.
Immediately liked the contrasting colors and title of click maths page.
definitly gonna do this on next math test
Great!!!
Beautiful
Blackredpen I admire your mathematical abilities!
Great explanation as always. As a request would you please do a video explaining the Pythagorean Theorem proof that used Trigonometry and Calculus on a "Waffle cone" diagram.
Very good
My favourite used to be the one I learned in linear algebra which involved vectors and inner products. However, I gotta say after watching this, I prefer this proof.
My favorite PT proof has for a long time, been President Garfield's proof.
My most unfavorite is the one from 9th grade geometry class. It was long, involved, and arcane. I don't even recall how it went any more.
I think it might have been right out of Euclid's _Elements._
Another one I rather like, sets the RT on c as its base, actually draws the squares on all 3 sides, then drops a perpendicular from the apex all the way down through the c square, and proceeds to equate each resulting rectangle with the square on the corresponding leg.
I do really like the one you've given, along with the inverse-squares version.
BTW, speaking of Pythagorean Theorem "cousins," the most remarkable one I've run across so far, involves areas of faces of a tri-right triangular pyramid (where 3 right angles meet at one vertex, so that those 3 faces are all right triangles). This then, is a 3D analog to the 2D right triangle. It can be made by a planar slice through any 3 vertices of a rectangular prism, no 2 of which share an edge - so that the 3 lines connecting pairs of them, are all face diagonals.
I'm having some trouble recalling the actual theorem at the moment; I must look it up; but I think it was just that the squares of the areas of those 3 RT's equals the square of the area of the 4th face.
I also seem to recall that it doesn't generalize into dimensions > 3.
In any case, have you seen that?
Have you perchance already done a video on it, that I missed?
Fred
I use the method of taking a square of side length a+b and drawing lines to create four triangles with base and height a and b respectively, with a square section of side length c in the middle. Comparison of areas shows the required identity
Cool!
What a delightfully elegant proof
Elliott Manley thank you!!
I've used inverse Pythagorean theorem to find distances between point and line in cube and other polyhedra without using a single coordinate/vector at all, it was such refreshing and satisfying way to find it that way without linear algebra.
So good
Mega fresh and retro funk!
No offense man, but I like the visual proof more. It's so cool.
Like the most beautiful equation(Euler's law) It's the most beautiful proof 👌👌👍👍👍👏👏👏Fantastic
This is how I have been taught to demonstrate the theorem at school when I was 10. It is the easiest, most intuitive way in my opinion. It is a natural consequence of Euclid's theorems on triangles, so it fully makes sense.
Good video
J M truthful good 😊
This guy makes me happy for no reason
Magnifique !!!!
Geniuas im impressed