Our Discord hit 10K members! 🎉 Meet students and ask top educators your questions.Join Here! Numerade Educator

# In mathematics, the Pythagorean theorem, also known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The theorem can be written as an equation relating the lengths of the sides a, b and c, often called the "Pythagorean equation": a^2+b^2+c^2. where c represents the length of the hypotenuse and a and b represent the lengths of the triangle's other two sides.

#### Topics

No Related Subtopics

### Discussion

You must be signed in to discuss.
##### Top Educators ##### Catherine R.

Missouri State University ##### Heather Z.

Oregon State University ### Recommended Quiz

#### Geometry

Create your own quiz or take a quiz that has been automatically generated based on what you have been learning. Expose yourself to new questions and test your abilities with different levels of difficulty.

### Video Transcript

you guys were back with another video. We're gonna talk with the Pythagorean theorem today. What Mawr historical or well known math topic is there, uh, maybe the quadratic formula. But if you ask anybody what they remember from math, Pythagorean theorem is one of those things. So we're gonna talk about again. The Pythagorean theorem. Now the storm is named after Pythagoras, the famous Greek mathematician. Although the idea behind the Farum was known 2000 years beforehand by the Egyptians. So it's not necessarily that Pythagoras was the original prove er or discover of the the're, um It's just that sometimes through history, through his teachings or some of his students, because Pythagoras was the leader of a mathematical cult, not in the negative sense that we think about today. But he basically had a bunch of followers that probably one of his followers kind of made it a big deal and attributed what they found to Pythagoras. At any rate, here's what the Pythagorean Theorem basically says if you have a right triangle and it's got to be a right triangle, specifically a right triangle, and I labeled the legs A and B in the high pot noose. See? Then if you square the legs and add them together, then you get the square of my partners. That's the classic Pythagorean theorem. So how can we improve this? Well, take this square, for instance. And what we're gonna do is we're gonna take a point on each side, the same distance away from each Vertex. So go there. I'm gonna go roughly there. Someone do my best to draw this out roughly there and roughly there. I wanna do some to connect them and what we're gonna get is this basically a square? Because if I'm setting that point or picking that point, that's the same distance away from each protects of the square, we will get another square that's approved for another day. But what I'm gonna do is I'm gonna label each of these triangle sides as a be and see So that would make this this'll leg a this leg B. Pardon me, this leg be and this hip hot new C and so forth, so I could go around the whole square and label everything accordingly. So if you think about the big square, the entire side length is a plus B, which means the area of the square would be a plus B times a plus B because it's a rectangle. So you're gonna do side time side that happen to be the same. Well, okay, that's a squared plus to a B plus B squared. So that's a expression for the area of the bigger square. Okay, Now what I want to talk about is the area of the triangles and the smaller square that I formed based upon where I picked those points. Someone get rid of this. Okay, So if we look at the inner square, the square here, we know that has an area of sea time, see, or in other words, C squared. If we look at one of these triangles that has an area of one half a bee which would be one half base times height Well, there's four of them. So if I multiply one half a bee times four, then we get to Maybe so. In other words, the sum of all of the four triangles and the smaller square is going to be C squared, plus two a b. But we already found the area of the bigger square, and that was found by this expression. So I'm gonna set a squared plus to a B plus B squared equal to C squared plus to a B. And then what happens? Thes cancel out and we're left with a squared plus B squared equals C squared. That's the Pythagorean theorem. So here's something. You have a right triangle. You know that this site is three, and the site is for and I'm curious about what the iPod News is here. Well, you're in theory. I'm gonna say, Take your one leg squared, plus your other leg squared and set it equal to the high pot news squared. That's nine plus 16 equals eight squared that's 25 equals h squared. And if I square root both sides, we get five equals H. That's one example of how the figurine theorem works. You could also go backwards if I Let's see, I gave you the high part news in the leg. You could find the site, so as long as I gave you two of the three sides, you can go ahead and find the missing side. So that is basically the Pythagorean theorem, specifically to a right triangle. And it's basically saying the sum of the squares of the legs equals the square of the iPod knees. See you next time Webster University

#### Topics

Rigid Motions (Isometries)

Volume

Terminology

Relationships Within Triangles

##### Top Geometry Educators ##### Catherine R.

Missouri State University ##### Heather Z.

Oregon State University 