 The name of Lecture 1 in our lecture series is the Pythagorean equation. What is the Pythagorean equation and why it's so significant? Well, the Pythagorean equation, or sometimes called the Pythagorean theorem, was named, of course, after the School of Pythagoras, which was a Greek school of geometers in ancient Greek times, for which they studied geometry in a very rigorous manner, and not in the Corbyn geometry system we'll be talking about in this lecture series, but because of the School of Pythagoras, the Pythagorean equation is named after said school. And so the Pythagorean equation is an equation about triangles, specifically about right triangles. So you see a picture of a triangle over here, so if we have an ABC triangle, a common convention we're going to do when we talk about triangles is that the side opposite of an angle will typically use the same alphabetic letter to describe the measure of this side and the measure of this angle. So we often will use a capital letter to describe the angle, the vertex of the triangle, and we'll often use a lowercase letter to represent the measure of the side opposite that. So you see there's a capital A for the vertex A, and then there's a lowcase A that measures the length of the side that's opposite the vertex A. So you see that convention. That's not just something we do for right triangles, that's something we do for triangles in general. So if you have a right triangle, if ABC is a right triangle, this happens if and only if the sum of the squares of two sides of the triangle are equal to the length of the square of the other side. So what do you mean by if and only if, right? So what I'm saying is if you have a right triangle, then it turns out that if you take the two legs of the triangle, which you can see right here, if you square those lengths and add them together, that's going to equal the square of the other side of the triangle, the hypotenuse. That's if it is a right triangle. But it turns out that if the three sides of a triangle satisfy this Pythagorean equation, the sum of squares of two sides equal the square of the third side, then that triangle necessarily has to be a right triangle. So let's see why the Pythagorean theorem holds in general. So this is going to be basically a proof by illustration. So take a look at this picture right here. So imagine we have a square for which, which you can see here in black, a square which each side of the square has length C. All right. And then what we're going to do is we're going to draw right triangles around the square so that the hypotenuse of the right triangle we drew. And so this is the right triangle ABC, the hypotenuse of that right triangle coincide with the length of the square. And so let's orient the triangles in this manner right here. So like if you play Ring Around the Rosie, you're going to get all four of these triangles. So I want us to consider the area of this problem in two different ways. Because of the way that we've oriented these triangles, it turns out that we've actually created a larger square that you see right here where the length of one of the sides is going to be B plus A or A plus B. The order doesn't really matter there. And so if this is a square, so one side is A plus B, this second side is A plus B, all the sides are A plus B. If we look for the area of that large square, it's going to be A plus B squared. That's how you find the area of a square. You just square its side. All right. But on the other hand, if you take this black square, which its side length is C, the area of that square is going to be C squared. And then if you look at the area of any of these triangles right here, it's a triangle so the area is going to be one half base times height. As it's a right triangle, the base would be B, the height would be A, or however you want to look at it, the perspective doesn't matter so much. But you're going to get four of these. So you get one, two, three, four. And so the area of this region should be C squared plus four times one half A times B. And so since we're calculating the area of the exact same region, these two areas are going to equate with each other. So A plus B squared is equal to C squared plus four times one half A times B. So let's then simplify this equation algebraically. If you take A plus B squared and you foil it, that is this is A plus B times A plus B. If you foil it out, you're going to get A squared. You're going to get A times B. You'll get another A times B and you'll get a B squared altogether. You foil it out like that. On the right-hand side, we'll just take four times one half, which gives you two. So you'll notice that on the left-hand side, we have A squared plus two AB plus B squared. On the right-hand side, you get C squared plus two AB. If you subtract two AB from both sides, you then get the Pythagorean equation. So using a little bit, basically using the area of formulas of rectangles, that is squares just being a special case of rectangles, and a right triangle is just half, of course, of a rectangle. If you take areas of a rectangle, you can very easily prove this Pythagorean equation analytically. Let me consider more of a geometric argument. Again, after all, the Pythagorean equations named after Pythagoras, who didn't use algebra to prove these things. And so these images you see on the screen right now are courtesy of Wikipedia. The idea is, if you were to consider on the left the picture we started with, four triangles and a square, a square whose area is C squared. On the right, though, you see that same square, but now we moved the right triangles so that the resulting white space, which is equal to the original white space, the area is A squared and B squared. So let me actually show you the animation. You can see that we start with the original configuration and then we move the triangles so that we can move things around so that the white space, which always had the same area throughout the entire journey, the white space started with C squared, but then the white area ends up as being A squared plus B squared. So again, that's a geometric argument to the theorem of Pythagoras if you prefer. All right, so let me show you some examples one can do with the Pythagorean equation. So let's suppose we have a right triangle ABC that you see on the screen. We know the hypotenuse of the right triangle is 13. We know that one of the legs is, well, we don't know it actually, it's a variable, let's call it X. But what we do know about this triangle is that it's hypotenuse is 13 and that one of the legs is exactly seven units of length longer than the other. What can we do with that information? Well, using the Pythagorean equation, this side squared plus this side squared because those are the legs will equal this side squared. So we end up with X squared plus X plus seven squared is equal to 13. We can set up a Pythagorean relationship between the sides of these triangles. And now this is a quadratic equation we're going to try to solve for X. So to begin, we're going to foil out the X plus seven squared. And as a reminder, this mean when I say foil out, I mean, this is X plus seven times X plus seven 13 squared, of course, is 169. And so when you foil that out, you're going to end up with an X squared plus a seven X plus a seven X plus a 49. This still equals 169. Combine like terms, you add the X squares together, you end up with a two X squared. Add the X's together, those are like terms, you end up with a 14 X. And then we're going to subtract the 169 from both sides because this is a standard technique when solving a quadratic equation. You're going to subtract 169 from both sides. You end up with a negative 60, which you could factor out the two. Now you're going to get two times X squared plus seven X minus 30 is equal to zero. And then if we continue to factor, we can look for factors of negative 30 that add up to be seven. And that happens when you take plus 12 and minus five. 12 times negative five is equal to negative 30, 12 minus five is equal to seven. And so we see here that one side, when you put, you set each of these factors equal zero, X could be negative 12 or X could equal five. Which negative 12 doesn't really make sense for the length of a side. You can have a negative length. Therefore, the side of the triangle needed to be five. But then you'll notice that five plus seven, oh, that's 12, that's kind of where the negative 12 came from. And so we see that we have a triangle here where one side is length five, one side is length 12 and one side is length 13. In the literature, this is often referred to as a Pythagorean triple. That is, this is a right triangle for which all sides of the triangle are whole numbers. And so five, 12, 13 is a Pythagorean triple. That is, if you plug, if you take five squared plus 12 squared, that's equal to 13 squared. Another very famous, probably the most famous Pythagorean triple is three, four, five. If you take three squared plus four squared, that's equal to five squared. And therefore there exists a right triangle whose side lengths are three, four, five respectively.