Easiest Proof of Pythagoras' Theorem

Proof of Pythagoras' Theorem using just basic areas and algebra in less than a minute!
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[tonitalas]

Very clear proof of pythagorian theorem! Thank you 😊

[Bedoroski]

Thank you. This is the most intuitive proof, no need to move the shapes around.

[Phymacss]

Your videos are really interesting and nice! Also, thank you for the kangaroo challenge practice course !

[AF-rukk]

The best! But how can we prove they're congruent ??

[rafiqueahamad]

Thanks man just have an exam and saw you really helpfull

[shantamajukar]

Really enjoyed bro
Love from India 🇮🇳 to you❤🎉

[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

[jirifrantal]

How do you know that the shape inside the outer square is a square (with right angles)? You need to prove it. ;-)

[MeganArmstrong-hp]

Where does the 2ab come from when finding the area of the larger square?

[mythologicaleditor]

But how will you prove that the insife rhombus is a square with side c?

[samyogdhakal]

I've a doubt. Why we make bigger square a square having length a+b not a rectangle with
Side 2a and 2b and still area of smaller square will be c^2

[cazanid]

Can i ask why theres only (a+b)2 why not 4?, and also where u can find that 1/2ab,

[adamgalambos]

I got seasick of the flickering text 😢

[stu]

The problem with the proof is that it's not obvious why the sides of the large square are all a+b

[MeganArmstrong-hp]

Where does the 2ab come from when finding the area of the larger square?