 Hi, I'm Zor. Welcome to Unizor Education. I would like to continue talking about certain properties of pyramids. And in particular, it's another lecture about the problems. Well, I wanted to present certain problems which are not like ordinary. Now the ordinary problem is, well, this is the pyramid. This is its base. It has such and such area. This is an altitude. Find the value, for instance. It's not really interesting. You just multiply times one-third and that would be the value. So I need something which is a little bit more involved as far as creativity, maybe some interesting approach, etc. So the problems are not really very difficult, which I'm presenting here. But they do require a certain turn, if you wish. So if you turn, then you will be on the right spot. Alright, so here is the problem. We have a triangular pyramid and there is nothing special about this pyramid. We don't know that this is something, some edges are equal or not equal. Angles are completely free kind of a pyramid, whatever can come. Now, what we do with this pyramid is the following. On the side h s a, we choose a point a prime. Some point on b, on s b, which is b prime. And let's say on c, c prime. Now, what I do know is how these points divide the edges. Basically, I know the ratio. a a prime divided by a prime s is equal to some coefficient k a. b b prime divided by b prime s is k b. And c c prime to c prime s is k c. I know these coefficients. So basically this is the ratio of this segment length to this or this to this and this to that. The next thing which I'm doing is I'm cutting my pyramid by a plane which goes through these three points. So I'm cutting in half. My question is how the volume of the bottom relates to the volume of the top. So can I determine based on the ratio of the edges the ratio of the volumes? And by the way, on the top I still have a pyramid. On the bottom I have completely some kind of a polyhedron, whatever you call it. So it's not the pyramid, it's not the prism, it's not something familiar. Let's put it this way. And I still would like to know how the volume of the bottom relates to the volume of the top. Okay, so here is how I can do it. First of all, I'll do exactly the same trick I did in the problem number one. I turned the pyramid on the side and I will consider s a b as a base and c as a top. And you will see why. So if I will turn it this way, I would like to know first of all how the area of s a b relates to the area of s a prime b prime. That would be the ratio of their bases in this case. And next, how the altitude of the big pyramid with an apex at c relates to this top part, top pyramid which is c prime. That's an apex of a small pyramid. So I would like to basically have the ratio of the bases and the ratio of the altitudes. And if I know that, I will just multiply these ratios and I will get the ratio of the volume. Okay, so first of all, let's talk about s a b triangle. I mean the area of s a b. How it's related to the area of s a prime b prime. That's what I would like to know. And for this reason, let me just view this as a plane geometry problem. So let me just draw this picture again. So I have some triangle. I have two points which divide in some ratio. So this to this is k a and this to this is k b. Now, question is how the areas are related. Now, the area of a triangle is a product of its base times its altitude and times one half, right? So if I consider this to be a base and this to be an altitude in big triangle and this to be the base and this an altitude in a small triangle, let's call this h prime and h. So what I can say is that the area of s a b is s a times b h times one half. And the area of s a prime b prime is s a prime times b prime h prime times one half. So let me basically find out how these two are related. Well, s a and s a prime, I know how they are related. Now if a prime a relates to s a s k a, s a relates to s a prime as s a prime plus a prime a relates to s a prime, right? s a is a sum of this plus this. Now if I divide it, it would be one plus k a. So I know this ratio, this to this. How about these two guys? Well, but let's think about it this way. Triangles s b prime h and s b h are obviously similar, right? Because these are two perpendicular to the same line, which means these angles are the same and this is the common angle. So they are similar, which means everything is proportional, which means b prime h prime relates to b h, s s b prime relates to s b, right? And this we know basically this ratio. So let me just write it this way. So what I need is b h relates to b prime h prime as b h to b prime h as s b to s b prime equals. s b is sum of these two, right? So it's s b prime plus b prime b divided by s b prime, which is s b prime is one and b prime to s b prime is k b plus k b. So this is the ratio between these two things. So I know that this is greater than this by this factor. This is greater than this by this factor, which means this greater than this area to area as a multiplication of these two, right? So the area of s a b relates to area of s a prime b prime as one plus k a times one plus k b. So that's how we know the ratio of these areas. Okay. Now about the volume. Actually, the story of the volume is very much similar to this one because the altitudes, if you think about this, is really behaving exactly like these altitudes. Altitudes of the pyramids are exactly these altitudes of these two triangles at the base. Let's think about it. s a b is the base. Now we have a perpendicular from c to this base. So let's have a plane, which is the base. Somewhere here we have this triangle s a b, right? Now somewhere here we have a point c, which is connected to all three of them. And somewhere here we have a point c prime. So we have dropped the perpendicular from c. That's an altitude of the big pyramid from here to this plane s a b. And we have dropped here the perpendicular from c prime to this base, which is exactly the same plane. It's still s a b plane because a prime and b prime are within that plane. So it's the same plane. So somewhere here we have b prime and a prime. Doesn't really matter. But what's matter is that these triangles s, I'll use the same letter h, h and h prime, both c, h and sc prime, h prime are obviously similar. Because again this is one straight line sc. And sh is projection of this line onto the plane, which means every line, every point projects onto the same sh. So considering these two perpendiculars, all the angles are equal, these angles. And within this plane, s, c, h, everything is within this plane. So obviously there is a proportionality. So c, h relates to c prime, h prime exactly as sc relates to sc prime. Which is sc is sc prime plus c prime c divided by sc prime, which is 1 plus kc. You see it's exactly the similar situation in a 3G as we had in the two-dimensional plane when we were talking about altitude within this triangle. Now that allows me, that allows me to basically compare the volume, the volume of the big pyramid, which is one-third area of s, a, b times c, h, right? And c, h is the perpendicular from here to here, and this is perpendicular from here to here. This is h prime. Now h and h prime belongs to s, a, b, right? So these two altitudes are proportional to this number. c, h to c prime h prime. Now we can go to the volume. So the volume of the big pyramid, which is c, s, a, b, divided by the volume of c prime, s, a prime, b prime. What is it? Well, this is one-third times this area, so one-third times area of s, a, b, times c, h times c, h, divided by one-third area of s, a prime, b prime, right? This one times c prime, h prime. Now one-third is reduced. This ratio is this, and this ratio, s, h to s prime, h prime, is this. So what's the result? The result is 1 plus k, a, k, b, k, c. Now this is the volume of the entire pyramid as it is related to the volume of the small pyramid. Now what I'm asking is what's the ratio of the bottom part of it? But let's just think about it. The volume of the entire pyramid, which is c, a, c, s, c, s, a, b, is equal to volume of c prime, s, a prime, b prime, a small pyramid, plus volume of some kind of polyhedron at the bottom, right? So if I will divide it by volume of c prime, s, a prime, b prime, that would be what? Now it will be this divided by the same thing, which is 1 plus ratio, which we need. Ratio of this divided by the volume of the small pyramid. So it's bottom part divided, which is basically x, right? That's what will be as a result. This divided by this is 1. This divided by this is x, whatever we need. So from here, considering that I know that this ratio is this, I can find out what's my x. That's the ratio between the bottom and the top, which is 1 plus k a, 1 plus k b, 1 plus k c, minus 1. So that's the ratio between two parts. Well, I don't know, it's not really very difficult problem, but what's interesting about this is the following. First of all, what we did, we turned it on the side, basically having these three lines like controlling the pyramid. These two control the base, and this one controls the altitude. And now, considering I know by how much we squeeze each part, I calculated by how much we squeeze the area of the base and by how much we squeeze the altitude. So that's also very important. And, well, by the way, incidentally, we have proved that for a two-dimensional case, this is exactly similar. So if you have some triangle and you squeezed one line, it's in certain ratio, k a and another k v, and have another thing. So the area of this triangle would be some similar formula we actually came up with the formula. So the big area divided by small area is 1 plus k a times 1 plus k b. So that's just like a byproduct of this thing. And this volume is just three components. So everything seems to be very much similar in two-dimensional, in three-dimensional case. Well, what I suggest you to do is try to do it exactly the same thing. Just try to make a nice drawing, derive whatever I have here just by yourself without looking at the lecture. Try. I think it's a very good exercise. It will actually help you to view things better because right now you do need some three-dimensional eye, so to speak. If you just turn this around, which one is the base, which one is the altitude, it's a good exercise. Well, that's it for today. Thank you very much and good luck.