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# Pythagorean theorem proof using similarity

Proof of the Pythagorean Theorem using similarity. Created by Sal Khan.

## Want to join the conversation?

• Does the Pythagorean theorem work for acute or obtuse triangles?
• The Pythagorean Theorem is just a special case of another deeper theorem from Trigonometry called the Law of Cosines

c^2 = a^2 + b^2 -2*a*b*cos(C) where C is the angle opposite to the long side 'c'. When C = pi/2 (or 90 degrees if you insist) cos(90) = 0 and the term containing the cosine vanishes.
• Is there any way for the hypotenuse in a right triangle to not be the longest side?
• No, there is not.
The lengths of the sides of a triangle go in the same order as the angles across from them :
the biggest side is across from the biggest angle
the medium side is across from the medium angle
the smallest side is across from the smallest angle

This also means that if 2 or 3 angles are the same, the sides across from them will have the same length. You see this in isosceles and equilateral triangles.

So, if we know that the longest side has to be across from the biggest angle, and a triangle has a right angle, the other angles cannot be bigger than 90, since the angles must add up to 180 total. The other 2 angles must TOTAL 90, so each must be smaller than 90. This means that their sides will have lengths shorter than the hypotenuse.
• can you explain why we can add together the following, and what the meaning of this addition is?

a^2 = cd
+ b^2 = ce

I understand the proof, but the addition of the two equations is confusing to me.
• OK, so you pretend you have any two equations:

3+5=8
4+2=6

I can combine all the numbers that are on the left side into one expression, and all the numbers on the right side into one expression. So we would have:

3+5+4+2=8+6
14=14

It makes sense, right? Sal did the same thing to the equations:

a^2+b^2=cd+ce

Hope this helps! If it doesn't, let me know ;)
• At , he uses the ~ . Does that mean anything?
• 5 years late, but if anyone wants to know, the ~ sign with the = under it means that two (or more shapes) are congruent to each other.
• I am a bit confused. Can someone please explain what Sal did?
• Sometimes its easier if you cant understand to go to the settings tab on the right hand side and watch the video again this time slower and with captions on. i highly recommend this other then just watching a different video not related to Khan Academy bc when you do that its a different type of formula and plan this is exactly how u get stuck on a problem. Before seeking any other kind of video or formula just try watching this video slower and with subtitles
• I understand the proof but why is it algebraically valid to add the 2 sides of the 2 equations?
a^2 = c * d
b^2 = c * e
a^2 + b^2 = c * d + c * e
These 2 equations don't seem to have any kind of relationship...
• Lets see if this helps by doing something simpler.
If we have a=b and c=d, then the equivalent would be a+c=b+d. So if we start with a=b, we can add the same thing to both sides and it stays the same. So a+c=b+c. Then, if c=d, we can substitute in to get a+c=b+d.
We could do the same on this by saying a^2 +b^2=c*d + b^2 (adding b^2 to both sides) and end up with a^2 + b^2=c*d + c*e (substituting c*e for b^2).
• Can anyone simplify what he just said? Please? I'm confused....
• Basically, what Sal was trying to say was that the sum of the squares of each of the legs of the right triangle was equal to the square of the hypotenuse.
• This is a general question about . . How do you prove that all the triangles are similar in the first place? I understand that he used the angles and color-coding but how do you do this with actual writing?
• Why would we add the two ratios ( on the video)?