Занятие 13. 5 класс базовый Геометрия: Площади, периметры, разрезания

Hello everyone! Let's start slowly. I hope that the board is visible and even that I am writing something is visible. Yes. Super, thank you. So, today we have a topic, well, I would say something like geometry, if you know this word. But a little more specifically, today we will talk about perimeter and area.

Most likely you guys have heard these words in the context of some rectangles and squares. And you probably even remember that to calculate the perimeter of a rectangle, you need to multiply 2 by the sum of the length and width. The square is probably the same, just the square is a rectangle whose length and width are the same. And the area to calculate the rectangle needs to be simply multiplied by the length and width. In general, of course, the facts are correct.

Although, if you know only their information and nothing else, then, of course, there is no understanding of the perimeter and the area. In general, I think that it is also not a secret for anyone that if we talk abstractly about the perimeter of some figure, then what is it? This is the total length of its border.

Of course, the perimeter is not only a rectangle, a square, but any figure that has a border, which can be calculated. For example, a triangle, a circle, or an circumference. All this has a perimeter. Just to calculate the perimeter of a rectangle, in this case, because it has the same length and width, you can use that form.

And about the area too. It's good to understand that it's not just a number that is considered by some formula, by some principle, but it's a certain number that is also in the figure on the plane. Let's say that it's in all of them. And it shows approximately how much space this figure takes. If the figure takes a lot of space on the plane, then it has more space.

And indeed, to find out the area of its square, it is enough to multiply by the number of numbers. These were all the definitions that I hope are familiar to everyone, or some properties or intuitive understanding. We will talk about something more detailed, as a volume, only in space.

We'll talk about it in more detail, how to solve different tasks related to it. Because it would be good to know how to calculate the perimeter of a rectangle or a square. But it would also be good to understand how these knowledge can be applied to solving some tasks that are more complicated at first glance. And in fact, nothing is more complicated than what we just discussed.

Well, let's look at some task for warm-up. Let's see how you are doing with the perimeters. And let's look at this task. So, there is a square. This is a square, if I draw. And it has such a rectangular cut. Let's do it like this. The cut is 2 rectangles. And we know that the perimeter of the first rectangle is 17 cm, and the perimeter of the second

13 centimeters. That's all we know, but we need to find the square side. Here. What conditions? Once again, there was a square. It was cut into such linear cuts into two rectangles. One has a 17-perimeter and the other has a 13-perimeter. And you need to find the square side. Well, you know that, okay? But as always, try to write me your thoughts on the square side.

But it's better to send something like a solution or an explanation besides the answers. Because the answer is good, but it's not clear how you solved the problem. So, I've received two answers so far. Two different answers. Oh, three different answers already. Interesting. Of course, two of them are wrong, at least. So, Roman, firstly, it's bad to do this.

Let's delete this message. Because... Oh, no, stop. So, Proman, delete the message. Don't write "Global Chat" anymore. Moreover, the solution is actually wrongly written there, if anything. But in general, if you deleted it, that's good. If you don't have it. But don't write "Global Chat" anymore. So, so far, by the way, I've only seen one correct solution.

So be careful. While you're still thinking, I'll start with this. I'm sure that some difficulties may arise related to the fact that you tried to understand what sides should be of these rectangles, if there are 13 and 17 perimeters.

It will be difficult to understand which sides, because the perimeter is an odd number. We already know that the perimeter is 2 multiplied by the sum of the sides.

If it was 18 and 12, it would be easier. 18, the sum of the length and width of the left is 9, the sum of the length and width of the right is 6. And somehow come up with something, choose. Here is one very important nuance. You may really want to try to guess the sides when you see the perimeter or when you see the area. Let's say the area is 20 square centimeters.

For example, you were told that a rectangle has a diameter of 20 square centimeters. And you are like: "Ah, 20, what is it? Is it 5x4, 2x10 or 1x20?" Well, there are three options, one of them will suit us. You shouldn't do that.

Because in fact, if the task is not stated directly in the condition that the numbers should be whole, long sides, then it can be very easy that the side, for example, will not be whole, say 10 cm and 5 mm.

It doesn't interfere with anything. So here too. But it's clear that the sides are not whole. Otherwise the sum would be even. But at least by selection, today you shouldn't try to decide. I see what you write to me. Vladimir, no, wrong. Yakov, right. Vlad too. We'll discuss it now. We are asked by Timofey, the side of the square, one.

Well, so far, in fact, the guys are more likely to be wrong, that is, they write the wrong solution. Everyone is wrong. Well, let's understand why. That is, many people write the answer 5 centimeters. Square root of the square. Well, you are certainly more likely to be right. And those who write 6, for example, are more likely to be wrong. Let's figure out what's going on. Look, many people add 17 and 13.

What will it turn out to be? To find out, let's try to tell by color what will happen if we add 17 to 13. 17 is the sum of lengths of all sides of the left rectangle. Therefore, when we add, it will turn out something like this. I'll mark it with green. Okay, when we add 13, we add another length. Now let's figure out what we've calculated.

We can see that we calculated the entire border, and something else was inside. What was inside? Look, we have this little piece in the middle, but it was not calculated once, as some of you may have thought. It was calculated twice, because it is in both the left and right perimeter. It turns out that when we add 17 and 13,

We calculated the entire perimeter of the square and again calculated the piece inside. And it is equal to the side of the square. It turns out that 17 + 13 is 6 sides of the square. 6 sides of the square. What is it? Well, it doesn't matter. 6 sides of the square. And what does it mean? It means that one side of the square is 30 divided by 6, that is, 5 centimeters. That is, sometimes it really

Sometimes it's really useful to add up the parameters you have in order to see what you will get in total. And maybe calculate it in another way, that is, by dividing it into other figures. Let's try to practice this technique a little more. Sorry, can you repeat it? How did we find it again? 17 + 13 is clear, but how to solve 6 square?

6 sides of a square. As we understood. But once again, here we are on the picture, highlighted in green, that the perimeter, well, highlighted in green is what is included in the perimeter of the left rectangle. Here we are, we have 13, we have folded the perimeter of green and yellow. And look at the segments that we have calculated. Here is green, what goes to the left, yellow goes to the right.

And now we look at it differently and say that the whole square border, the red one I highlighted, we calculated once, when we put it together. The part from the green and the part from the yellow. The whole square border turned out. These are four sides of the square. There is also an inner segment in the middle. Once we calculated it in the green part, once in the yellow part.

And it's equal to the side of the square. So, we've calculated the perimeter, which is 4, and two more, left and right. So, it's 6 in total. — Thank you. — Got it? — Yeah. — Super. And let's practice a little more in this, and solve the problem, but not about the square, the rectangle, but about the triangles. Look, here's a triangle, and another triangle in the center. Draw it like this to divide our whole triangle by 4.

Let's say that the perimeter of a large triangle is 20 cm, and the perimeter of small triangles is 9 cm, 10 cm, 11 cm. And you are offered to find the perimeter of the central triangle based on these data. The large triangle has a perimeter of 20 cm, so we divided it into 4 triangles, the three extreme have a perimeter of 9, 10, 11 cm. What is the perimeter of the central triangle?

You can also write me a personal message. So, Artem is right. Yakov, it's not very clear what it is on the left. In the sense, it's not very good when negative numbers arise in a decision. Because it doesn't make any sense. So that you could really look and understand. Aha, well, one on the left, one on the right. Therefore, they are of course the same. It doesn't work like that. So, so, so. Yes, but everyone writes.

Let me wait a little more. Yes, everyone writes an answer. They would also say why it is like that. They just said that the answer is like that. But oh well. Well, yes, everyone writes an answer of 10. In general, probably, if you understood how we solved the previous task, then this task should not cause any serious problems. Let's try to do the same. That is, let's see. Well, we have perimeter of triangles: 10, 9, 10 and 11.

Let's fold the perimeter of the extreme triangles and see what we get when we do it. Let's fold it. I'll take the green one. Yes, it's 9, 10, and 11. I folded three perimeters: 9, 10, and 11. And what did I get? Well, we can look at it differently and say that I calculated the entire border of the triangle, that is, the perimeter of the large triangle.

And what's left is what's left inside. There's a small triangle inside. So, yes, it turns out that this 30 centimeters that we got is 20 centimeters plus what we need to find. Well, from here, of course, it follows that what we need to find is also 10 centimeters of the perimeter, like the upper triangle. Great. Let's move on to the next task.

Let's quickly try to answer this question. It's not even a task, but rather a observation. Excuse me, I don't understand how such a task is solved. Okay, let's figure it out. So what did we do? We have folded 9, 10 and 11. That is, we have folded the perimeters of the length of the sides of the extreme triangles. So now it's clear.

In case it's better to ask at the moment when it becomes unclear. We just bypassed the green one, what we have made. And now we look at what we really got in total. What is highlighted in green. We got the entire perimeter of the large triangle. Because we calculated that the entire border of the large triangle is highlighted in green.

And what was inside? Inside was the perimeter of this triangle, about which we need to understand something. And it turns out that when we folded 9, 10 and 11, we got the sum... Well, we got 30, right? This is 30. And this is the sum of the perimeter of the large triangle and the small triangle, which is in the center. Because we just explained why it is so. Because it turns out that the sum of the borders...

large and small. Well, since the perimeter of the large is 20, it turns out that 20 plus something is 30, then something is 10. This is how we found the perimeter of the central triangle. I understand everything, thank you. Super. Well done asking. This is important. If you don't understand something, it's better to ask.

So, the third question, or rather the third task, or rather the exercise, as you understand it well. Look, let me draw a grid for clarity. May I ask a question? No, let's draw a grid for you. Yes, you may ask a question. No, that's it. I understand. Okay.

Look, here is a square. And now I will draw another line with a blue color. Look. Oops, sorry, I erased everything. Let's see something like this. Here is a square, and there is such a blue thing inside it. In any case, all these sections are directed in the same way as the sides of the square. That is, all the corners are straight.

You can look at it and send me a message. How do you think, which figure has a larger perimeter? The square or the blue one? Whose perimeter is larger? You can count it intuitively. Wow.

Oh, phew. I got all three options right. I thought everyone would get it right. Of course. Round. Yeah, I got all three options right. Someone said that... Why three? Someone said that the square is bigger, that the white one is bigger. Someone said that the blue one is bigger. And someone said that they are equal. The blue one, the blue one. Well, it's not necessary to turn on the microphone now. It's better not to do it like that.

Well, yes, the options are different, of course, but one prevails. In general, of course, in fact, the guys write that they have the same perimeter. Although it may seem that, well, how is the square, it is so straight, and therefore, probably, all the pieces are small, and in the blue we go like this, zigzag, so it will probably be longer. But in fact...

Here, of course, to make everything completely correct, it would be necessary to do it like this. Although, in general, it doesn't matter how to do it. It could be like this, for example. And it would still be the same. In fact, the answer is that they have the same perimeter. Let's understand why. Look.

Let's look at this particular section. Look, what happens if I move this section here? I'll just move it. And I'll move this section here.

Do you agree that the perimeter of this blue thing that we have slightly green, it will not change? That is, once again, I will take, I will erase this thing, leave pixels, some big pixel erased, I will erase these things and just move them. This perimeter will not change.

I just moved them. But look, I have more of them on the square border. But now what prevents me from doing this, for example? Yes. Oops, again, I erased it. Of course, it should be like this. Like this, for example. I also moved it, and nothing changed. But now I can do the same. Do like this.

And nothing changes. Do you agree? Well, obviously, I can do this until the end and in the end completely remove all the internal segments to the border. And it turns out that they actually have the same perimeter, although you can be deceived and really think that the blue one is bigger or the white one is bigger. But those who originally wrote that they are equal, of course, you will not spend so easily. They are actually equal.

Super. Well, now let's finally look at some task that will be more complicated. And the task will be like this. Danil, why did you turn on the microphone now? Will you tell us? Only it's not clear. But if you interfere, you'll have to delete the lecture. Then you'll see the recording. So it's better not to turn on the microphone. If I don't ask you separately.

So, what is the task? We have a rectangle, oddly enough, and we will cut it into 4 smaller rectangles with two linear cuts. And we will know about some perimeters. Let's say this perimeter will be 22 centimeters long, and this one will be 24 centimeters long. The question is, respectively,

What is the perimeter of the entire rectangle? This is in point A. And there will be point B. For those who can handle the point. Here in point B, we will have a perimeter of 23 centimeters. This is still unknown in point A. And the question is, what is the perimeter of the fourth rectangle? You can think. But this is point B. Point A is just four rectangles with two opposites, which are the left top and the right bottom. The perimeter is unknown. You need to find the perimeter of the entire rectangle.

So, again, try to write not only answers, but also some thoughts. I mean, why do you think the answer is like that? Some explanations. Relatively to the line. Well, a rectangle or a square with a perimeter of 22. How does it work? He has a line that is below him, and it goes...

Like the upper line of the 24th perimeter. Does it go straight like that? Yes, yes, yes. This is all rectangular sections that are in the same direction as the original rectangle. That's it. Thank you. So... Oh, oh, oh. Why are they sending me answers today? So, guys, can I still ask those who send answers to explain at least a little bit where you get this answer from? That is, why does what you write give us the correct answer at all? Where does it come from?

or incorrect answer, the answer you got, then I want to understand where they come from, your answers. So, Vlad wrote. Well, it's about clear. Let's do this. I'll look at what you send for some time now, and then we'll ask someone and he'll tell us. This person, whom we'll ask, what can be done. For now, in the "A" first. Maybe even show.

I'll erase the B point for now, then we'll come back to it. So far, we have already written the answer. Let's do this. You can raise your hand if you think you can tell the solution to this problem. So that everyone understands. Let's see. Here, for example, Artem Shcherbakov quickly sent the answer.

Artem, can you tell us why you got this answer? It's better to show it, because I've enabled the ability to comment on this board, so you can draw something on the board with some color. But guys, let Artem do it. Yes, I can. I'm ready. Come on, come on. So, we, similarly to the previous task, do it like this. Look.

Here we have the perimeter of these rectangles. We will now compare them with our large rectangle. For example, this side can be moved here, and this side, this side, and this side. So, all the sides of these two squares

Oh, sorry, these rectangles will be on the sides of our large rectangle. This means that their perimeter, well, the sum of their perimeters is equal to... 22 plus 24 equals 46. Uh-huh. Yes. Good. Good, Artem. Thank you very much. Yes. Let's then... Now...

Let's remove this option for now. We'll have to erase all the comments now. How to do it now? Well, okay. In principle, it's okay for now. Well, yes. As Artem said correctly, you can just take and move the relevant segments to the border of the rectangle. By the way, I don't know, I'm just curious.

But maybe someone had such thoughts. In such tasks, which are related to the perimeter, it is sometimes more convenient to look not at the entire perimeter of the rectangle, but at its halves. What do I mean? For example, if we look at the first task, let me draw it again for now. The first task in which we took a square and divided it into two rectangles with 17 and 13 perimeters. In fact, it was possible to solve it as another. Now only

We will delete all comments. How else could we solve it?

And we could only look at the half-perimeters. That is, not to take the length and width twice, but only once. What would be better in this case? It would be better that we would not have to make a mistake with how many times the internal pieces are considered. Because nothing would be applied to each other. And if we look only at the halves, then it turns out that you can take the length and width once here, at 17.

And you can take the length and width once here at 13. And thus get only three sides of the square. And that's for sure, nothing is laid here, nothing is crossed. It turns out that 17 plus 13 is half. It turns out 15 is three sides of the square. Three sides. And something similar could be done here. Although not necessarily, of course. Namely, if you take just all

That is, the entire perimeter. It turns out that we have some parts both on the border and inside. And we need to figure out how to transfer these parts. And if we take only the half-perimeters, then we will also have to transfer. But maybe we will have to transfer less. Because we can take only the part that is on the border of the big one. We took it.

Now it's not very clear what to do, but now we can do it like this. That is, not four pieces, but two. In fact, the same solution, just a little bit with a different approach, which is based on what we take exactly on the perimeter, and not the perimeter itself. It also turns out that if we fold halves, then we get halves. Therefore, if we fold whole, we get whole. But that's it. It was just a lyrical digression. Now let's discuss point B, which we still had here.

We were given that the perimeter of this one is 23 cm, and it remains to understand how to calculate the perimeter of the remaining square. So, are there any... maybe someone who would like to tell how to do it quickly? Maybe Yakov Leonov will tell us. Ready? Yes, I can. Come on.

Can we turn it on? Well, how to draw? What? Yes, why? I think you can probably just say why the answer will be like that. Or do you need to draw? Okay. It turns out that Artem gave an example when we translate the sides of the rectangle by 24 and 22 centimeters. Well,

on the other side of the big rectangle. Then the rectangle, which is 23 centimeters, is two sides of the rectangle 24 centimeters side and top and bottom side of the rectangle 22 centimeters.

And the rectangle that is under the question has the same thing. Its extreme sides are the sides of a 22 cm rectangle, and the upper and lower sides are the sides of a 24 cm rectangle. Then we understand that the blue, which is marked by a blue color, their perimeter is equal to the rectangle,

a rectangle marked in white. Then 22 + 24 = 46. 46 - 23 = 23. Then the answer is B = 23. Thank you very much, Yakov. In general, what did he do? Let me explain, otherwise it might not have been clear. We look at such a perimeter, at the 23 perimeter, and say that this side can be

For example, this side can be moved here. And the second one can be moved here. And the second one can be moved to the sides of the white rectangles. And the sum will be the same. But in general, it would be good to solve the problem if we solved it from scratch. But before that, we solved the problem with point A.

So I would expect that looking at the task in point B, you can just click your fingers and say, "Oh, well, here's the answer 23, that's why." Is there anyone who can also say the solution of the task in this format? That is, it is very, very easy to explain why the answer is actually 23.

So, Vladimir Stryzhkov, can you explain why the answer is 23? Without going through the segments. Yes, I can. Because we have 22 and 24 there. We found out that the perimeter is 46.

And it turns out that in these two rectangles, well, that is, in these two diagonally, we will also have, if we fold them, there will also be a perimeter of a rectangle. But we already learned that it is 46, so we need to subtract 26 from 46, and we get 23.

Of course. We didn't have to solve the same problem again, we already transferred all the segments we needed. In this case, we found out that if we add the left, top, right, bottom, i.e. two opposite rectangles of their perimeter, then we will get the perimeter of the entire rectangle. And how is the left, top, right, bottom pair different from the left, top, bottom, right, top pair? In fact, they are not. They are also opposite rectangles. Therefore, if one has a sum of the perimeter, then the other has a sum of the perimeter exactly according to the same idea.

Well, if the number of perimeters is a large perimeter. And so it turns out that 24, that is, they have white numbers, and blue ones are the same, so 23 should be blue. Super. Thank you very much. Well, we practiced with the perimeters. Now it's time to move on to the squares. But before that, let's solve a transition task with you. Which will immediately make us think about the perimeter and the square. Look, the task will be as follows.

There was a rectangle. What did we do with it? One side was increased by 3 times, and the other side was reduced by 2 times. We got a square. What does the square side equal to? If it is known that the original rectangle's area is 54 square meters. So, there was a rectangle. What did we do? One side was increased by 3 times, the other side was reduced by 2 times.

twice. It turned out a square. Find its side if the area of the original rectangle is 54 square meters. Well, let's measure. If the area of the original rectangle is 54 square meters. Well, that is, yes, what happened approximately. There was some kind of rectangle, we increased one side three times, the other one was reduced by two times. It turned out a square. I can't get a square.

Is it longer? Something like that. Yeah. Oh, Nikolay is typing something. Let's see. Yeah, Artyom is right. "So, Yakov, it's not true that you wrote this." What? What is this action? I don't understand. I mean, I wrote the truth, but what is written on the left? I don't understand. "So, please write how you got the answer."

Yakov, I got it. How many of these actions are there? What are they? How? I don't get it. Just in case, I'll say that the fact that the square of the rectangle is 54 square meters doesn't mean that you can choose between the options. What if he has sides 6 and 9, what if he has sides 3 and 18, what if he has sides 1 and 54? Because I've been saying this from the beginning, right?

that it's not a fact that the sides must be whole. That's the first thing. And the second thing is that even if you could find the answer, that is, say that the sides were like this, then everything seems to match, then you will not solve the problem yet. You will just give some example. That I managed to find the dimensions, that if the side was like this, then it seems that it fits the condition of the problem. You don't answer the question at all, you don't even bother. And what if there are some other options?

So if the task is not said in reverse, then if there are several options, then you need to find all of them. If there are two options, let's say, such squares that could turn out, then suddenly you will not find one of the options. It turns out that you have not completely solved the task. Therefore, it is better not to try to go over.

but to come up with some solution that would tell us that we have already figured out all the options. So, Alexander Mironov answered correctly. Let's see what we have been given. We have been given the area of the original rectangle and what happens with the sides. We have increased one side three times, then reduced the other one by two times. If you still have difficulties with solving the problem, think about

How does it affect the square? How do we change the direction of the rectangle? So, Ivan, I also didn't understand what kind of actions are these. Why are we doing this? To get an answer. Because we first divide... No, no, no, wait, wait, wait, wait, wait. We haven't discussed the decision yet, so it's better not to answer in private.

To not spoil the possibility of solving tasks for everyone who has not done it yet. But so far, they have not sent so many solutions, by the way. Right. There are two or three things I saw. No, by the way, Yakov, it doesn't work like that. Well, if something, then something. But that's not true. So, and Kirill Makarevich, what is the second action? And what is the first action? I don't understand yet. So, let me tell you slowly.

You can do it all with a picture. You had a rectangle at first. I have it marked in white. And it has a square of 54 square meters. Then we did something with it, and it became a green square. You can do it step by step. For example, first we could reduce one side of it, and then increase the other side, for example.

Think about what happened with the area every time. That is, how much space this figure occupies on the plane. That is, after the first operation, when we squeezed, then when we stretched it to the square. What happened to the area? If you try to answer this question, you will find out the square area. But the square is a good figure in this sense, because if you know the square area, then, unlike the square with a rectangle, you are, in principle, able to understand what side of this square should be.

So, Nikita Palfyrov, this is more of an answer. Yes, this option is possible. And are there other options? This is more of a question as to why there are no other options. So, what, did you all calculate the square? Look, here you have the square, right, at the rectangle. What is it? It's just a product of its two sides. Yes, length to width. Here. And now, yes, now imagine that you had some sides, let's say, at the rectangle. And then you took it and increased one of the sides by three times.

Or let's first reduce it by 2 times. It doesn't matter how you do it. There was a rectangle, you reduced it by 2 times. What about the area? Look, the area is the product of two multiples, length and width. One of the multiples remained unchanged, and the other was reduced by 2 times. That is, it divided by 2. What is left of the product of two numbers? Let's say, a/b. You took it, and a is the same, and b is divided by 2.

Of course, it's the same as doing this. So, all the work has also been reduced by 2 times. If one multiplier remains unchanged, the second one is reduced by 2 times. And it turns out that the area, when we reduced the side by 2 times, the area also reduced by 2 times. Okay, so what now? If we took 54 square meters, we reduced one side by 2 times, then the area became like this. And then we

One side was increased by 3 times. And one side remained unchanged, and the second side was increased by 3 times. And it turns out that the area multiplied by 3. And we got nothing else than the area of a square, which is equal to 81 square meters.

But now it's not enough. If you have a square with a area of 81 square meters, then you can try to choose such a number, it is only one, which in the product itself gives 81. It is clear that it is only one, because if the number is larger, the product will be larger, if the number is smaller, the product will be smaller. So if we choose such a number, we will win. But, probably, from the multiplication table it is easy to understand that 81 is 9 by 9. And it turns out that the side of the square is 9 meters.

The square is 9 meters. And now you can see what the original rectangle was. It became 9x9, but at the same time one side is three times larger, and the other side is two times smaller.

So one side of 9 divided by 3 is 3, the other side of 9 multiplied by 2 is 18. It turned out that the original rectangle has a size of 3x18. But you couldn't think about it until you solved the problem. Because 3x18 is suitable, it turns out a square. What if it turns out another square with other dimensions of the original rectangle? This is also an important part of solving the problem. If it's not clear. What was not clear?

Why do we divide 54 by 2 and then multiply by 3? Well, okay. The whole stage of transformation is clear. We had a white rectangle in the picture, and now we have a green square. So how can you turn a white rectangle into a green square?

We have two sides. We reduced one of them by 2 times. Here you can see the one that is long. And then we increased the short one by 3 times. And we can do it sequentially. Of course, it takes time, but we can do it sequentially. The rectangle will still turn into a square.

Okay, now we look at what happens with the area during each operation. We had a big rectangle, white. We left one side unchanged, this one, and the other one was reduced by 2 times. In fact, we cut off such a part from it. Now the side has become 2 times smaller. What happened to the area? In general, it is probably intuitive to understand that it has become 2 times smaller.

Well, the fact that the place has become twice smaller. We cut off the same one. But we can explain this from the point of view of the square formula, which is the division of length by width. Well, we have two multipliers. Length multiplied by width. Let me write it like this: length multiplied by width. This is the square. And what did we do? We did not change the width of one multiplier in any way, it remained the same.

And the first one, which is length, it decreased by 2 times. That is, length divided by 2, multiplied by width. It became a new area. But if we reduce one of the multipliers by 2 times, then all the product will also be reduced by 2 times. Because, if you remember, it is not so important in what order to divide and multiply, you can rearrange them. It turns out that here is length multiplied by width.

Divide by 2. The same thing. So, the reduction of length by 2 times, or one of the multiples by 2 times of the length, also reduces by 2 times. Is it clear so far? Yes. Okay. And we look at what happens to the area. In the first action, we divided 54 by 2. And it turns out that after the first step, when we reduced the length by 2 times, our area became 27 square meters. It was 54, and it became 2 times less. It became 27.

And then we did the same thing, but we increased the other side by 3 times. And here is the same thing, if one side remained unchanged, and the other side increased by 3 times, then the area of the work also increased by 3 times. And then there was an area of 27, we increased the side by 3 times, and the area became 81. And thus we just calculated the area of the final square, which was obtained after these two changes in the length of the rectangle. This is also clear so far, right?

Yes. Well, if we know that the square of 81 is a square meter, then we can easily find its sides. Well, how easy? We already know that the square of a square is the division of a side by itself, because the length and width are the same, they are the same number. Well, what number does the division of itself give 81? Only 9. 9 times 9 is 81, and if the number is greater than 9, it will be greater than 81. If it is less than 9, it will be less than 81. Therefore, 81. Therefore, 9.

So, the square side is 9 meters. That's what we wanted to find. All clear, thanks. Super. Well, okay, we figured out the answer. And now let's... Oh, sorry. Let's remember that before that we took a rectangle and divided it into 4 parts. Let's do the same thing. Well, there were perimeters, but there will be areas. Look. Take a rectangle. We divide it into 4 parts with such rectangular cuts.

And we will know not the perimeter of these parts, but the area. Let's say the area is here, and the letter "S" is usually indicated. Here is the English word "square". 18 square meters. This part has a 24 square meters area. This part has a 27 square meters area. That's what's given there.

Of course, you probably guessed that you need to find the area of this part, the remaining fourth rectangle. Try to do it. And just in case, I'll remind you again that it's not necessary that the numbers are whole. In other words, that the length of the side is a whole number of meters. Yes, you can certainly try to sort out 27, what is multiplied by what, 18, what is multiplied by what. Thus, if you manage to pick up the numbers, you will only be able to find out the answer.

Maybe it's not so bad, because knowing the answer, you can solve the problem, prove it entirely, it can be easier. So, in general, if you want, you can try. Damn, it's hard. You can try to choose numbers, and you will succeed. What lengths of these sides of the rectangle can have such areas? And so you will probably be able to get some kind of answer. If in the example you choose, what will be the area of the upper left? It's even interesting, you will succeed.

So, Vlad, did you pick this one or did you decide differently? Well, yes, actually, I said that it wouldn't be a full-fledged solution. Oh, no, Yakov, it doesn't work like that, of course. If it's the first one you sent. It won't be a full-fledged solution. But in general, the idea of trying to pick up numbers is not bad here, at least to understand the answer. You will not prove it yet. I mean, you will just pick up some example that

Well, it's not the only fact. There are other options, but you need to find all the options. Any math task, if not the opposite. If there are more than one option. So you don't solve the whole task. But at least you can pick an example. First, practice the skills of selection. And secondly, with a ready answer, knowing what to prove, it will be easier to come up with a solution. Danil, is there something related to the number you sent?

Well, the fact that they all share it, probably yes. So, Alexey, you picked this one, right? Wow. Timofey, can you tell us why you made such an expression? Well, tell us in writing, that is, not in a personal message. It certainly ... Well, the answer turned out to be incorrect, but the expression is interesting. I wonder how it turned out. So, Kirill, no. In fact, yes, it should be schematic.

It's similar to the picture, but schematically, I mean, compared to other rectangles. It's true that Leo the Upper has the smallest area among all of them. So if you have more, it's probably not so. So, Roman. It's true that all the figures have an unknown area. But...

It turns out that there is only one option, which is possible with these three square values and three rectangles. Only one. Because there are not only the properties that you wrote, but also other images that connect them. And in order for all images to be completed, in short, for them to be completed, any one is not suitable.

No, Vasya, it's a strange number. Okay, I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see. I see

I have 27. 27 is a number that is not represented in the form of two multiples. It's either 3x9, but there are no other options, because we can't take 1x27 if we think that the sides are whole, because 27 is already a lot.

So, it's probably 3x9. So, if this rectangle has whole sides, one side is 3 meters, the other side is 9 meters. So, which side is 9 and which side is 3? Well, here you can look at 18, which is 9x2. Well, it's by ratio. For example, 27 on top of the... Wait, wait, wait, wait, wait, Roman, Roman, wait, wait, wait, wait.

I just don't know what to say, but if you suddenly want to say something that can be solved by the task, then you shouldn't do it. But in general, yes, you can just notice something here. 9x3, but at the same time 24x9 is not divided, and 18 is divided. So, most likely, 18 is divided by 9, and 24 is divided by 3. 24 is 3x8, and 18 is 2x9. It turns out that according to our assumptions, this side is 9, this side is 3.

This side is 3, this side is 8, and this side is 2. So, it's 9, 2, and 8. If you try to choose the answer, this option is possible. 8x2 = 16. I remind you that we haven't solved the problem yet.

But we have a very accurate example that works. In which we have those three squares that we were given. And in this example the answer is 16. Okay. Let me tell you a secret. You guessed it yourself. 16 is the correct answer. There are no other answers. Only 16 can be the answer. But why? What if we don't try to guess and choose the right ones?

We will try to solve the problem entirely. How do we act in this case? So, now, Tissimophey, look, I'm not talking about where you got this solution, I'm more about why this solution works. So, did you write it, just some kind of expression?

I hope you understood that there was an error in this expression. But it doesn't matter. After we showed the answer. Sorry, I forgot that I had a unit at the beginning and forgot to finish it. I understand. But I'm talking about why the expression you wrote, let's not speak about it, why it leads to a bad answer.

Can you try to explain it? Not in written form, but not yet. I didn't understand it myself. I just saw that there was such a method. I didn't solve the problem, I just decided to look at the solution and saw that there was such a method and took note of it. I see. In the future, such a advice is better in mathematics.

Try to understand what you see and what you use later. My sister also said that this formula will always work with push-ups. I just want to say that there are

In general, you study mathematics, and it is clear that mathematics is not the simplest science, and there are things that are really difficult. There are things that you will hear and immediately do not understand. Why does it work? But you will really want to believe it. You will say that it seems to work, and they will use it. It seems to be true, but why is it true? What's the difference?

Well, for studying mathematics, this approach is not very good. Because...

There are many reasons why. One of them is that if you do this, you are not studying mathematics, you are applying it. But if someone asks you: "Why is it like this?" "Well, I was told so." "Okay, but what if it's different in this situation? How will it work?" "I don't know, I know how it works in this situation, but I can't tell you how it works in this one." So you are not a mathematician,

Well, a person who doesn't know mathematics, but uses it. There are a lot of people like that, of course, but it's not worth comparing yourself to them if you want to study mathematics.

I'm not saying that Timofey made a serious mistake. But if there are things you don't understand, you should ask why they are true and ask someone who can explain them to you. And understand them. But it's okay, Timofey. I hope you'll understand how it works today. There is also a funny anecdote about the mathematics of physicists who

They needed to go somewhere for the benefit and they bought tickets. But I don't know how appropriate it is to tell such anecdotes now. Maybe I'll tell you after the lecture. So, let's look at this task. Not that I want to spoil the solution, but I want to say that in fact,

And it's not like the perimeter, the area. It can't be solved in the same way. But this task is similar not only in terms of the wording, but also in terms of the solution. With one difference, that the topic is about the area, not the area. If the area is the sum of sides, then the area is the product of the sides. Therefore, we can try to mark the products that we consider in these areas.

Let me just make it a little bit clearer and write down what it means that the area of these three rectangles is 18, 27, 24. Let's look at the upper right rectangle. It has a square area of 18 square meters, and this is the definition of its length by width. Let's find some length on this figure, find the width, and just note what we will multiply in order to get 18.

Let's do it this way. I have two options for width. This segment and this segment. Which one do you think is better? Left or right? We need to multiply width once. Which one do you think is better? Left or right? Just your guess. Left. Left. Right. Right. Left. Left. Right. Left. Right. Left. Right. Left. Left. Right. Left. Left. Left. Right. Left. Left. Left. Right. Left. Left. Left. Left. Left. Right. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. Left. left. left. left. left. left. left

What can be useful for solving the problem? Well, of course, there is no real difference, but if I take the right one, then the right part is only a part of the upper right rectangle. And if I take the central part, then it is included in both the upper right rectangle and the upper left rectangle, which we need to know something about. From this point of view, it is probably more logical to take the central part. Danil, enough.

Okay, let it be highlighted as red, and this one will be highlighted with a blue part. For the same reason, we take the middle part. I will write that I have a red part, multiply by a blue part, and it turns out to be 18. I will not write the number, I am not humble. Okay.

What else do I know? Let's look at the lower right. There are 27. I already have one side, it's a blue segment, so let's designate the second side and write the same. That blue multiplied by green is 27 square meters. So far everything seems to be clear. We just write down that the square of a rectangle is the length multiplied by the width.

Okay, let's move on. Here we have the lower left. It has one green side, and one more side that we don't know yet. But we'll mark it now. It turns out yellow. And then it turns out that green multiplied by yellow is 24. And we have to understand what red divided by yellow is. So I rewrote the conditions. Well, now

Why did he start to separate these things into squares? Now what? Now, in general, let me do it this way and give you a couple of minutes to solve the problem after that. In fact, a recording in this form, it may seem to be nonsense in itself, but it is actually a pretty serious hint. Try to understand now how we can definitely find what we need.

Wait, maybe we should have picked some tasks today? Which ones?

Well, we were supposed to sort out four tasks from the seminar. And I'm embroidering something there.

Let me tell you that the tasks of the seminar are discussed and discussed. The other tasks are discussed at lectures, but not yet. Only one week has passed since the last lecture.

It would be too early to analyze the previous tasks. I remembered all the rules. What, Vladimir? It turns out that we will analyze these tasks for the next lecture. In the seminar, it seems.

Let's not talk about it now, you are somehow distracted. Let's deal with this task. It is important to understand what is happening and how it is being solved. Yes, Maksim, that's right. And Mikhail is wrong. So, Timofey, let's try to tell us how it works.

Well, we have to multiply 18 by 24, then we will know the length of all sides. Red, blue, green and yellow. And we will know the length of all sides. And then we have to... Wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait,

We will find out all the sides, their result. If we multiply all the sides. The product. Yes, the product. And then we will have to divide by 27 and remove two sides: blue and green. And we will get, and we will only have red and yellow. It will be equal to 16.

Yes, absolutely right. Look, since we multiply the area, not add it, it would be convenient to multiply these things. Because if we multiply 18 by 24, we get a product of a red line, but a blue, green, yellow one. That is, each one will be counted in this product with four multipliers. And if we multiply 27 by a question,

then we will also understand the product of the same color. Since the product of the variable of the place of multiplicity does not change, then we got the same thing. And this simply means that you can divide 18 by 24 by 27 in order to find the question. Because 18 by 24 and 27 by the question are the same thing. So, to find the question, we take, multiply, and we get 16. That is, 18 by 24 is 27 by 16. So, was it clear, guys?

How did you solve this problem? Yes! But if you have questions, you better ask them. Because if it wasn't clear to you, it would be like with Timofey, who heard about the solution, but didn't understand why it worked like that. Tell us the joke that you were told to tell at the end. I'll think about it. Maybe I'll tell you later. Okay, so far, we have questions. Any questions?

If you have, ask. If you don't understand something, it's important to understand the solution of this issue. It's very important. Then it will be very useful to you. Well, okay. If there are no questions, then let's finish.