 Hey everybody, welcome to Tutor Terrific. Today, I have a proof of the Pythagorean theorem which I call an alternate proof because I already have put up a proof on my channel using two squares, one inside the other. This proof is actually a lot simpler than that, and first I want to explain what the Pythagorean theorem is for those of you who are just stopping by. The Pythagorean theorem for right triangle says that the square of both legs added together, so the square of side A here next to the right angle plus the square of side B next to the right angle is equal to the square of the hypotenuse for a right triangle that's the side opposite the right angle. Those of you who've done geometry at all, you're very familiar, maybe even algebra, with this formula. Now, this proof makes use of a right triangle specifically, and I've labeled the sides as such, A, B, and C. Now, what you do for this triangle is you make the altitude from the hypotenuse. So I draw, I put the triangle on its side such that the hypotenuse is at the bottom, and I make an altitude which is a line that is perpendicular to one of the sides and extends to the vertex of the opposite angle. So this is a right angle here as well. Now, I'm going to label the sections of side C that are cut by this altitude right here. I'm going to label the short part here Z. So this section, I'll move this down, this section right here is Z. Then over here, this would be the leftovers of C, so I'd subtract Z from all of C, and I'd get this portion. I'm literally going to label that C minus Z. So now that I have that labeled, I'm going to make use of the fact that I've created three similar right triangles out of one. And this is what you do when you make the altitude from the hypotenuse to the right angle. You actually make three similar right triangles. Now, you have to remember what similar triangles mean. These mean triangles with the same angles but proportional sides. So it's like I took one triangle or one shape and dilated it with a certain scale factor, as you remember from if you've done transformations yet. There's something true about similar triangles that have the same angles and proportional sides. Well, just that that is true. Corresponding sides are proportional. That means I could set up proportions of corresponding sides, and that's what I'm going to do here. So this triangle here is the smallest of the three. This triangle here is the middle-sized triangle, and then the main triangle here is the largest right triangle. All three of those are similar to each other. I'm not going into that proof, but I'm assuming that's true for this proof. Okay, so now I'm going to set up some sides. Now, I did not label the altitude because it's not involved in any of my proportions. I'm going to look at the small triangle first, this one here. I'm going to take the shorter leg and the hypotenuse. I'm going to make a ratio of those two sides, z over a. Okay, now I'm going to look at and set that ratio equal to that same ratio of sides in the large triangle. Okay? That would be the short leg would be a for the big triangle and the hypotenuse would be c. So for the small triangle, it was short leg z over hypotenuse a and then the large triangle, that's short side a over hypotenuse c. Okay, so z over a would equal a over c. This is what some people refer to as baby equals mama type proportion. We've got the small sides in a ratio of the smaller figure on one side and the larger sides of the larger figure on the other side of your proportion. This is one proportion we are going to use. Now the other proportion involves the middle-sized triangle and the large triangle. Okay, I'm going to do the same thing here, but with the larger leg. I'm going to set up a ratio of the larger leg, which is c minus z over the hypotenuse b. So I have c minus z over b and I'm going to set that equal to the larger leg of the large triangle over its hypotenuse. That would be b over c like this. Okay, now that I have these proportions, I'm going to cross multiply. I'm going to multiply both sides by their respective denominators and so that these are no longer fractions. These denominators will move as the arrows point out. That's cross multiplication. That's one result you can get from cross multiplication. Over here on this side, I'm going to get that a squared when I multiply a times a equals c times z. Okay, and yes, I can reorder where those end up because of the commutative property or the associative property. a equals b is the same as b equals a. Alright. Over here, I have a little bit more complicated expression, but that's not a problem. I have that b squared equals, now this is c times c minus z and I can distribute that c into that binomial there, which would give me c squared minus c z. Now you see the three big players that we were trying to prove up here, a squared, b squared and c squared. So some intuitive person might say, hey, let me add this expression for a squared to this expression for b squared and that's exactly what we're going to do. So I'm going to say, I'm going to determine what a squared plus b squared is using these expressions. Well a squared is c z and b squared is c squared minus c z. So I'm going to add that to c z. Okay, so you might be confused until you notice that boom, c z minus c z is on the right hand side. That's going to cancel, which leaves you a squared plus b squared equals c squared, which is what we were trying to prove. We've just proven the Pythagorean theorem using similar triangles, a very simple method and a good one to know as an alternative to my other videos method, which again uses two squares, one inside the other. Alright guys, thanks for watching. This is Falconator signing out.