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Russian
00:00:02
n’t seen you for so long, although just00:00:04
last week we had a video with you00:00:06
that was about 8th grade algebra, to be00:00:09
honest, I looked at your00:00:11
comments and many people write that we00:00:14
spend a lot of time on basic00:00:16
mathematics, that is 7 8 9th grade and today00:00:19
we will do super and [ __ ] and talk about00:00:23
stereometry,00:00:24
that is, take popcorn and everyone who came00:00:27
to 10th grade, even those who probably came00:00:30
already to 11th grade, let's repeat what kind of00:00:32
thing this stereometry is00:00:35
[music]00:00:37
let's start by comparing What is the difference between00:00:40
this stereometry and the geometry that00:00:43
was in the seventh and ninth grades?00:00:44
Previously, you looked at everything in a plane,00:00:47
for example, in the plane of a notebook, we drew00:00:50
all sorts of triangles and squares there00:00:51
and we studied them in stereometry; we are already talking00:00:54
about three-dimensional figures, oh, here we live00:00:57
in three-dimensional space the00:00:59
same thing in stereometry there are some00:01:01
three-dimensional figures that have, for example,00:01:03
like a cube has length, width and height, but00:01:06
what is the problem is that we are not00:01:09
moving away from the notebook paper, we00:01:11
will not glue models to solve00:01:13
problems for example on cubes, we will still00:01:16
draw00:01:17
it on a plane and the most important thing in00:01:19
stereometry is to learn to include the00:01:21
imagination because it will00:01:23
imagine as if we are going around a00:01:25
three-dimensional figure being studied,00:01:27
we are drawing something inside, this is the main00:01:30
problem, and now let’s see how00:01:32
specific examples, here I have a00:01:34
cube drawn, what’s special here are the lines00:01:36
that are sort of behind the00:01:38
figure, they are drawn with a dotted line, and those00:01:41
lines that in front of us need to be drawn00:01:43
solid, and we need to somehow adjust00:01:45
the eyes so that we look at the figure and00:01:47
imagine what this is looks at00:01:49
us, but there’s a wall somewhere hidden there, the00:01:51
same thing is here, here we have a pyramid and00:01:53
these lines that are drawn are00:01:56
solid, we see them, and then that behind it is00:01:58
something invisible that they will00:02:00
offer us, they will write the name of00:02:03
some tasks figures and ideally you need to00:02:05
be able to work with each figure, understand what00:02:09
each figure is called, how to draw it,00:02:12
let’s quickly go through00:02:14
the figures that you need to know, a00:02:16
prism is a polyhedron whose two faces are00:02:19
equal and in the squares00:02:21
these will be the bases of the prism and the remaining n00:02:24
faces are a parallelogram and lateral the edges of the00:02:26
prism, then a figure appears: a straight00:02:30
prism is a prism whose side faces are00:02:32
rectangles a regular prism is a00:02:35
right prism at the base of which there will be a00:02:38
regular polygon, for example an00:02:39
equilateral triangle or a square, the00:02:41
next figure you need to know is a00:02:44
parallelepiped is a quadrangular00:02:46
prism at the base of which lies a00:02:48
parallelogram a right parallelepiped is a00:02:51
parallelepiped whose side faces are00:02:52
rectangles and at the base there is a00:02:54
parallelogram; another variety00:02:57
is a rectangular parallelepiped, this is a00:02:59
straight line guys boiling at the base of00:03:01
which there is a rectangle, that is, all the00:03:03
faces will be rectangles. The diagonal00:03:06
of a polyhedron is called a segment00:03:08
connecting two vertices of a polyhedron00:03:10
that do not belong to the same face, and00:03:12
then let’s figure it out with00:03:15
pyramids00:03:16
there is a polyhedron pyramid in which00:03:19
one of the faces is a polygon is the00:03:21
base and the remaining faces will be00:03:23
triangles with a common vertex00:03:24
and special pyramids are a regular00:03:27
regular pyramid, this is one in which a regular polygon lies at the00:03:29
base00:03:31
and the side edges are equal to00:03:32
each other, a triangular pyramid has a special name00:03:35
they call both a00:03:37
trader and the right traders00:03:40
call one whose all faces00:03:43
are regular equilateral triangles about three-dimensional00:03:46
figures, we talked about what we00:03:48
will rely on when we solve00:03:50
problems in stereometry,00:03:52
there are three axioms of stereometry, we will now00:03:54
look at them and then find out what00:03:58
consequences we need I will help00:04:00
again when solving problems,00:04:01
axiom number one, if two points of a line00:04:05
lie in some plane, then all the00:04:08
points that belong to this line00:04:10
will be in this plane,00:04:11
let’s try to draw it to make it00:04:13
clearer, but we have00:04:15
some kind of alpha plane and if some00:04:19
two points on the line lie in this00:04:23
alpha plane, then all the points that00:04:27
are on the line a.b.00:04:28
they will also belong to this00:04:30
plane alpha it seems that such a00:04:32
thing is obvious now let's move on00:04:35
what axiom 2 axiom 2 says that if00:04:38
two planes have a common point then they00:04:41
have a common straight line along which these00:04:43
planes intersect how could this be00:04:45
drawn00:04:46
for example here we have some kind of00:04:48
alpha plane, and let it be some kind of00:04:52
beta plane,00:04:53
so if they intersect these00:04:56
planes, then they00:04:59
will definitely have a common straight line along which they00:05:01
intersect,00:05:03
but you can do it somehow like this, that same00:05:05
straight line along which there is an intersection of these00:05:07
planes and the third axiom it says00:05:11
that through any three points that do not00:05:13
lie on the same line,00:05:15
only one plane can be drawn, for example, if00:05:18
we have points a b and c, then we can00:05:21
specify only one plane of the00:05:23
alpha plane these are the very three axioms00:05:25
on which we will rely, and what kind of00:05:27
corollary we will now deal with them,00:05:29
corollary number one, if we have a00:05:32
straight line and a point that does not lie on this00:05:35
straight line, then they define a single00:05:36
plane, at least some plane alpha,00:05:39
how can this be proven00:05:40
all the proofs go through the same00:05:43
axioms that we said before00:05:45
a little earlier, if we have a straight line, we00:05:48
can choose any two points on it,00:05:50
let’s say. 1 and . 2 and look what we00:05:54
have. . .00:05:56
Moreover, there are three points that do not lie on the00:05:59
same line, this pair lies, but with00:06:01
this third they are not on any straight line,00:06:02
which means that in axiom 3 they define a00:06:06
single plane, that same00:06:07
plane is alpha, consequence number two, if00:06:10
we have two intersecting lines and00:06:12
then they define again, the only00:06:16
plane, how this can be explained, we will00:06:19
again refer to the axiom if if00:06:21
the lines intersect, then there is a common one. and00:06:24
then we can choose, for example, one00:06:26
point from a straight line,00:06:27
let it be and the second point we can00:06:30
choose a straight line b00:06:32
and here in front of us are three points that do not00:06:34
lie on the same straight line ab, axiom 3, they00:06:36
define a single plane, the00:06:38
one we called alpha, now00:06:41
corollary number three00:06:42
if we have a pair of parallel lines,00:06:44
then they define a single plane,00:06:46
let out the plane alpha, as this can be00:06:48
proven, again we refer to the third axiom, we00:06:51
can choose00:06:53
two arbitrary points a and b on the first line,00:06:56
take another point on the second line, for example,00:06:58
point c and now we see three points00:07:01
of which they lie on the same straight line, which00:07:02
means they define 3 xiaomi a00:07:05
single plane which we00:07:06
called alpha, that’s the principle and that’s all00:07:09
when we solve problems on these if00:07:12
if we refer when we need to00:07:14
justify something, well and a little more about00:07:17
straight lines, if in geometry 7th 9th grade and we00:07:23
worked on a plane, then for two straight lines there00:07:24
were only two possible options for00:07:26
location; straight lines could either00:07:30
be parallel or they could00:07:32
intersect;00:07:33
if we work in space, then00:07:36
three situations are possible;00:07:38
straight lines can still be00:07:39
parallel they can intersect,00:07:42
this may happen, now00:07:44
I’ll try to draw on a plane, one00:07:46
straight line, for00:07:47
example, lies in the plane like this and the00:07:49
other straight line intersects this plane like this,00:07:52
pierces but with the first straight line00:07:56
and has no points in common, this00:08:00
case is called intersecting lines,00:08:02
how to understand that in general straight lines00:08:04
intersect; they do not lie in the same00:08:06
plane;00:08:08
straight line b also intersects the plane in00:08:12
which straight line a and this are located.00:08:15
is not on a straight line, but this can be00:08:18
written in the form of symbols; straight line00:08:19
a is contained in the alpha plane;00:08:23
straight line b intersects the alpha plane at00:08:28
some point a and we know that . but does00:08:33
not belong to a straight line, but this is a00:08:37
sign of intersecting straight lines, what else is00:08:40
interesting in stereometry at00:08:43
first, one of the most difficult tasks00:08:45
is constructing sections of00:08:46
polyhedra, so we will now look at00:08:49
such a typical case where it will00:08:51
construct a plane cross-section of a00:08:53
polyhedron passing through three points an00:08:56
hour that’s all I’ll quickly draw here our00:08:59
task, we have a pyramid with a b c and00:09:02
there are three points that lie on the edges of00:09:05
this pyramid,00:09:06
what they want from us is that you construct a00:09:08
section with a plane that will00:09:09
pass through these three points,00:09:12
this task is very scary for many guys, they don’t00:09:14
understand what it is at all they begin to00:09:15
simply connect these dots and in00:09:17
fact everything here is guided by literally00:09:19
one rule: you can connect only those00:09:21
points that lie in the same plane,00:09:23
turn on your imagination00:09:25
and find those points that can be00:09:28
connected right away at first. to00:09:31
and . m they lie in the plane of the bsk face00:09:35
in this triangle, which means without any problems00:09:38
we have the right to connect them like this, the00:09:40
MPK about is ready further and there are still points00:09:47
that can be connected00:09:48
yes m and n they both lie from below in this00:09:53
triangle a b c that is, from the bottom of this00:09:56
pyramid in we connect at the base, just00:09:59
keep in mind that this line00:10:01
passes from below, which means we don’t seem to00:10:04
see it, so we do it all using a00:10:06
dotted line about connected, and then here00:10:10
we are, it turns out that point n and00:10:13
point k cannot be connected because this00:10:15
line will go inside the figure a00:10:18
We don’t have the right to do this, as soon as we run out of00:10:21
pairs that can be connected,00:10:23
just start extending the lines and00:10:26
looking for some new points that00:10:28
will lie with the points you need in the00:10:31
same plane, for example this00:10:34
m.n. we have the right to extend there to the00:10:38
left side and the same straight line00:10:40
has neither beginning nor end, so we wanted to00:10:43
extend it, now let’s see what other00:10:45
lines we can extend, and those00:10:48
that also lie in the lower plane, so00:10:50
if we extend bc this is the00:10:53
line which lies below in the lower00:10:56
plane, then we will not get anything good;00:10:58
it will go there, moving away from00:11:01
Olya and will not go around moving away from the other00:11:03
side, that is, they will not intersect somewhere further, but00:11:04
if we take it and00:11:07
extend c and in this direction, then there00:11:10
This will appear below. .00:11:15
new example launcher. x why it’s00:11:17
good, it’s located on the lower plane,00:11:21
plus imagine if we00:11:27
pierced that back wall and sc this same plane in that direction, it would00:11:31
turn out to be something.00:11:32
x are still not only on the lower00:11:35
plane, as if extended to the left00:11:37
side, but also00:11:38
located on that back wall00:11:40
and the suit, which is also extended to the00:11:43
left side, this one. it is simultaneously00:11:45
in the lower plane and in the plane of the cotton wool00:11:49
here at the back, which we can see if00:11:51
we say so, we go around the figure, but this point is x00:11:55
and .00:11:57
if you look carefully they00:12:00
are just in the same plane, in that00:12:03
back one. k lies and stsy.00:12:06
x, as we said, lies in sc only00:12:08
extended to the left side, so feel free00:12:11
to now take x and connect it with .00:12:14
here we see it and then there will be a00:12:18
dotted line because this line moves along the00:12:21
cotton wool back wall so we connected point x and ka00:12:25
and now we understand that we have a00:12:28
new one like this.00:12:30
let it be. the game y and n also00:12:35
already lie on the same plane; the triangle00:12:38
was, which means we have the right to connect them00:12:41
about everything, now we see within the00:12:45
plane n y, how can we generally understand what a00:12:54
section by plane is, imagine the00:12:57
knife that we took and cut according to our00:13:00
figure, and so that the knife passed through the00:13:03
very points that we were told about in the00:13:05
condition, but the most important thing is that if you00:13:08
cut a figure with a knife, it’s as if from now and00:13:10
right through, and all the cuts should go along the00:13:14
walls of your figure, here we have00:13:17
a cut along this side wall00:13:20
and here that cut with a dotted line, he went00:13:23
there along the back wall to m along this00:13:26
front wall and m n here from below like this and this is how00:13:30
the section turns out again, what00:13:33
rule do we use to connect at once00:13:35
you can only connect those points that lie in the00:13:37
same plane those pairs of points if suddenly There are00:13:40
no such points anymore, so we are trying to00:13:42
extend the lines, extend the straight lines and00:13:45
look for some new points that00:13:47
will be in the same plane with00:13:49
those that exist, and to be honest, stereometry is such a00:13:52
thing that at first the body00:13:54
resists, so we will take00:13:56
more information in doses for00:13:59
today, I will not burden you Next00:14:01
time I promise that I will give 3 key tasks for00:14:05
which you can00:14:07
navigate well in this initial00:14:09
part of stereometry, but I00:14:12
have three interesting questions for you and if00:14:15
you manage to answer them, be00:14:17
sure to write it in the comments. The00:14:18
first question is why tripods00:14:22
cameras have only 3 legs, so it00:14:25
seems to you that this is absolutely not related00:14:26
to stereometry, but in fact, if you00:14:28
think about it or look at it again,00:14:30
review our lesson, then it’s quite possible00:14:32
that you’ll understand why everything happens exactly00:14:36
like this. Second question: 3 flies flew into the room through the window,00:14:40
what are the chances? out of00:14:42
a million that in exactly one minute they00:14:45
will be in the same plane and00:14:47
question number 3 is00:14:49
how to make four equal triangles using 6 matches00:14:53
write your answers00:14:55
in the comments I will definitely watch them00:14:58
and answer with a like if the answers are00:15:02
correct and we will finish for today00:15:04
if there was a video If you find it useful, be00:15:05
sure to like,00:15:07
subscribe to our channel, don’t forget to00:15:09
put the bell on to see00:15:11
the notification when our new video comes out,00:15:12
don’t forget about the first free00:15:14
lesson, and I’m off to prepare the next tasks,00:15:17
everyone, happy bye00:15:21
[music]Description:
😎 Выбирай для себя курс по математике с Ольгой Александровной. Первый урок БЕСПЛАТНО: https://www.tutoronline.ru//kursy-po-matematike ✔️Запишись к репетитору в школу TutorOnline: https://www.tutoronline.ru/repetitory ✔️Получи ответ на любой вопрос от профессионального преподавателя: https://otvetka.tutoronline.ru/ Часть 2: https://www.youtube.com/watch?v=te3yBvnP2pE Ваш любимый репетитор TutorOnline Ольга Александровна подготовила новый видеоурок по геометрии для 10 класса. И главная героиня этого урока - стереометрия. ► Жмите колокольчик, чтобы не пропустить новые видео! ► Не забывайте ставить лайки, друзья! Подписывайтесь на наш канал: https://www.youtube.com/user/RuTutorOnline Пробуйте бесплатный урок: https://www.tutoronline.ru Присоединяйтесь к нам в социальных сетях: ВКонтакте: https://vk.com/tutoronline Фейсбук: https://www.facebook.com/unsupportedbrowser Инстаграм:https://www.facebook.com/unsupportedbrowser
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