 Hi, I'm Zor. Welcome to Unisor Education. We will talk about triangular prisms today. This is part of advanced mathematics course offered on Unisor.com. And that's where I suggest you to watch this lecture from, from this website, because there are notes. Notes are very important. It's like a textbook for you. All right. Now, we have learned a lot about prisms. When we talked about parallel pipettes. So if you remember, parallel pipette is a prism with parallelogram as a base, which means the second base is also the same parallelogram, and every side face is also parallelogram, and then we were calculating the volume of this object. Well, I would like to continue talking about other prisms, in particular today's triangular prism, and about its volume. I will also use concepts of symmetry, which we have introduced in previous lectures. Symmetry is very important for this particular case, because that's how I decide to work with volume of the triangular prism. All right. So, what is triangular prism? Well, triangular prism is obviously the prism with triangle ABC as a directress. Obviously, we can talk about the right prism if the side edges are perpendicular to the base. But this is not important for this particular lecture. Actually, I think in the future, if you will deal with prisms, primarily it will be the right prisms when these edges on the sides are perpendicular to base, whether it's the prism with a square or a parallelogram as a base, or a triangle, or pentagon, or hexagon, or whatever else. So prisms usually in all the real problems are right prisms. But again, it's not a matter right now. We have introduced a concept of volume first for right rectangular prisms, if you remember. And then we calculated this as a product of the area of the base times altitude. That was how we did it for right rectangular prisms. Then we expanded to any parallel and basically the formula is still the same for any prisms. But that's what I'm going to do right now. I will talk about triangular prism and I will try to prove that this formula still holds for triangular prism. It's actually a simple thing to do if I will use the concept of a symmetry which I mentioned already. And here is how I would like to do it. From the plane geometry you understand that a triangle can be, using a central symmetry can be expanded into parallelogram. So if I know that the area of the parallelogram is length of the base times altitude using this transformation, which is actually a central symmetry relatively to, for instance in this case, relatively to this point. So this is the middle of this edge. So this point symmetrically transform into this and this is transferred into that because this is the central symmetry which means we have to draw a line through the center of symmetry which is on that line in the middle which means this point goes to this and this to that. This is obviously the line through the point let's call it A to A prime. Now, and this segment AP is equal to A prime P and it's very easy to prove that this is a parallelogram. And also since symmetry transforms an object, in this case a triangle into congruent triangle their areas are the same which means that the area of a triangle is just half of the parallelogram. So this is the idea for the plane geometry and I will use exactly the same idea for this case. I would like to construct another prism next to this one which touches this one using this same concept of central symmetry. So I will choose the line, I will choose this as a center of symmetry. B prime is a parallelogram because every prism has a parallelogram as a side. So this is the center of diagonals and let's see what happens if I will transform the whole triangle prism symmetrically using this as a center. But obviously B prime goes to C C goes to B prime, B goes to C prime, C prime goes to B. So these four points will be exactly where they are in the symmetrical object. Now points A and A prime will be different basically what happens is this not exactly the straight thing that's how it will be. So that's my new prism now this would be let's say D and this would be D prime so A will go to D and A prime will go to D and A will go to D prime. Now it's B, D, C or BCD, B prime, C prime, D prime is a prism. For obvious reason because if you remember that any line centrally symmetrical is reflected into line so AA1 will be reflected to D prime parallel to the original line so these are parallel and considering if you will connect A prime with D and A with D prime you will have basically parallelogram AA prime D D prime. Just because if you just consider this as a plane since these are parallel we have already proven that we can draw the plane. So within this plane it's very obvious that this is parallelogram. So it's very easy to do the proof in all the details which actually I don't want to spend right now the time because it's rather simple and again you want to send me your very detailed correct proof that this is exactly the same kind of a prism as the original. Well obviously it should be because all the segments are symmetrically reflected to similar segments with the same lengths and angles are preserved. Everything is preserved. So the volume should be double. So the volume of the A, B, D, C, A prime, B prime, D prime, C prime which is a parallel pipette is supposed to be double of the original triangular prism. And that's what actually makes me believe that the area of the parallelogram which is the volume of this parallel pipette is the area of the parallelogram at the base times altitude. Now altitude is the same for a triangular prism but the volume should be half but at the same time the area of the base for a triangular prism which is a triangle A, B, C is half of the parallelogram A, B, D, C. So the formula remains exactly the same for a triangular prism because volume should be half and the area of the base should be half but the H altitude is exactly the same. So that's the very, you know, kind of a schematic proof. It's not a real proof because it doesn't go through all the details. This is a very schematic proof that this is the formula for calculating the volume of the triangular prism. Well, basically that's all I wanted to say today about triangular prisms and obviously certain problems will be introduced a little bit later but this is just a theoretical material. We can always call the triangular prism right as I said before if the edges on the side, side edges are perpendicular to the base. Also if the base is the regular equilateral triangle it might be actually the prism based on this if triangle A, B, C is equilateral then the prism can be called the, how can it be called something like a regular or whatever, regular right prism. Now it's interesting actually that prisms are really used in optics just as a side issue because if the light goes through the prism in this direction for instance then it will be it will go through the prism and it will spread into a rainbow colors as you know because the white color has all these components and every components is changing direction differently this is the glass so the glass changes the direction differently based on the frequency of the light and the frequency of the blue side of the spectrum is, frequency is greater the wavelength is smaller so that's why we have this rainbow colors when it goes through the prism. So anyway there are some very important usages for triangle prisms in optics and probably some other cases as well alright so that's it for today, this is just an introduction into triangle prisms and the formula for the volume that's it for today, thank you very much