Proving the Pythagorean Theorem

We learned about the Pythagorean Theorem, but where did it come from? How do we know it's definitely true? What if old Pythag just made it up off the top of his mystical skull? Lucky for us, in math we can proof that things are definitely true, and there are tons of ways to prove that the Pythagorean theorem is true. Here's one of the simpler ones for you to memorize, so you can bust it out at a party and be the coolest kid in the room!

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[qbslug]

Short and easy. An even shorter proof involves drawing an altitude through the right angle which creates two right triangles and then using properties of similar triangles.

[CephasCarter]

Thank you Dave so much! You have been a lifesaver for my Physics class and now for my Geometry class.

[gilber]

You can also do it by rearranging the 4 triangles to make two smaller squares, with side lengths of A and B, and since the vacant spaces in both figures have equal area, QED

[mathiasjohansen]

Short and easy - made us feel warm and fuzzy - you the best Dave! <3

[alvisinger]

Most intuitive and easiest to understand, awesome

[HonestADVexplorer]

I didn't know mathematics could make me smile :)
Thank you, Dave! <3

[A.K-U.D]

thanks for explaining your video really helped me a lot in my maths project.

[Who.isbrian]

I finally get it, you're awesome!

[IshanKumar-ju]

That was quick and easy thank you so much ❤😊

[erictorbet]

How do we know the inner square is a square? Is it because we know the two acute angles of a triangle add up to 90 degrees?

[BuleriaChk]

The group operation of addition requires that two elements exist

Every number is prime relative to its own base: n = n(n/n) = n(1_n) Every even number is the sum of two primes (Goldbach): n + n = 2n. A number cannot be both even and odd.

Fermat's Last Theorem is valid.
Proof for Vilage Idiots
# = a + b
#^n = [a^n + b^n] + f(a, b, n) (Binomial Exansion) proved by Newon.
#^n = [a^n + b^n] iff f(a, b, n) = 0
#^n <> [a^n + b^n] QED

a=4, b=3
# = 4 + 3 = 7
#^2 = 49 = [25] +[24]

Note that for n = 2
#^2 = [a^2 + b^2] + [2ab] (Pythagoras and Einstein and anyone who uses vector calculas is wrong because of [2ab]

c= a + ib, c* = a-ib
cc* = a^2 + b^2 using imaginary numbers. But all numbers are positive???
cc* <> #^2
Note that #^2 = [cc*] + [2ab] imaginary numbers not necessary in full expansion.

[OseiDaniel-zi]

Trust me u are doing a great job sir 👍

[srishtisinha]

Pls make video on permutations and combination

[lelestrelinha]

THANKS, I was crazy because I have a work to do on the Pythagorean Theorem, I didn't understand it until I saw your video, thank you very much

[andonrangelov]

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?

[The_Green_Man_OAP]

Try this:

a²+b²=(a+b)²–2ab

Consider: c²+2ab
Factoring:

The value 2ab/c² is the double angle
trig ratio sin2A, (see ‡ at the bottom)

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:

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

[briannewton]

I always simply took the theorem as read, I have used it so many times. It is so cool to see the proof. I doubt a flerther would drop by, but I hope that if they did, no-one would hold you responsible for their brain exploding.

[gamerzblueduck]

I want to ask why it must 4 triangle to proving the theorem? Is it any other example to proving it with just one triangle?

[DrBell-slet]

It makes no sense that geometry is rushed in public schooling. They just feed you a^2+b^2=c^2 without explanation.

[samia]

okay but this actually made me feel warm and fuzzy