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Russian
00:00:00
hello dear friends, tanir boev is here00:00:03
and in this video I will tell you all the geometry00:00:07
which you will need a new one for00:00:09
mathematics00:00:10
this is what you studied in the textbook00:00:14
Atanasyan 7th to 9th grade00:00:17
many people don't like geometry because it doesn't00:00:19
understand00:00:21
they say that this is because of the lessons00:00:23
there is less geometry but I don’t agree with that00:00:26
in fact, each of you is capable00:00:29
the main thing is to understand and love this subject00:00:32
convey it clearly and clearly for00:00:36
one horn is unlikely to work, but I00:00:39
I'll try I'll try for you00:00:42
we were able to do it really well00:00:43
prepare for g I prepared for you00:00:46
a file that contains all the tasks that00:00:49
legs meet in mathematics and they00:00:51
located in a convenient location for studying00:00:53
ok00:00:54
this file has been tested by time and thousands00:00:58
my students that's why it works00:01:02
print and use the link below00:01:05
description for this video00:01:08
now before I start I'll tell you00:01:11
another news is open right now00:01:15
enrollment for our course where we prepare with00:01:18
students00:01:19
in 7 days we complete all tasks00:01:24
we analyze all tasks in clear language00:01:25
who will be on the exam00:01:27
that is, literally the day before00:01:29
exam and solve everything you need00:01:30
meet you then go and with00:01:32
write this exam with peace of mind00:01:35
so we knit it all within 700:01:39
days for five hours online00:01:41
records of each rock are preserved therefore00:01:43
you can review them whenever you want00:01:45
you want to00:01:47
each lesson in your personal account on the website00:01:50
Omsk st you will have homework check this00:01:54
homework and you will be able to contact00:01:56
questions that arise for you but suddenly00:01:59
after lessons you can do something unclear00:02:02
ask questions 24/7 and receive00:02:05
response almost instantly within 500:02:07
minutes00:02:08
also to get you started with my lessons00:02:11
because the lessons themselves are just beginning00:02:13
May 12 so that before this date you can00:02:18
I'm doing something useful to pass the time for you00:02:20
prepared a video course and task number00:02:23
1-5 we will also go through them informatively00:02:26
to be able to just look00:02:28
in advance I will make videos00:02:30
seven types by number by number 1, tier 500:02:32
these are all seven types that will be found on00:02:34
coals and task number 14 by the way00:02:37
statistics in this problem most often00:02:40
schoolchildren make mistakes so I make mistakes too00:02:43
added here write atanasyan this is the author00:02:49
in a geometry textbook, write this word a00:02:51
there in the message our groups00:02:53
VKontakte and we will answer you and tell you00:02:56
what is the tariff and how much does it cost?00:02:58
what is the schedule00:03:00
claim we will finally enroll 8 in this course and00:03:03
you can have peace of mind after the patina00:03:04
exam because you exceeded me00:03:07
all all all tasks that's the story yes00:03:11
I highly recommend a good course every year00:03:13
we have a lot of good reviews all over00:03:16
I like it if I if it’s not possible00:03:20
yes, at least just take it before00:03:23
exam in a few days and try00:03:25
about solving all problems00:03:27
ok or at least repeat what00:03:29
learning Dotera it helps a lot00:03:33
points00:03:34
I'll say this right away00:03:37
this course will give guys if you have for example00:03:40
level two or three then you can00:03:42
after these seven days until four and00:03:44
fives00:03:45
give me this is not some unfounded00:03:49
promises are real these are results00:03:51
confirmed by our students last year00:03:54
ok I mean it's absolutely real00:03:56
if you try, of course if you00:03:58
don't try, just sign up00:04:00
that we can’t see any results, so let’s start00:04:03
all geometry00:04:05
9 classes and so the street and types of corners are necessary00:04:10
to start with this there is an acute angle less00:04:13
90 degrees is a right angle, it is exactly00:04:17
90 degrees00:04:19
there are obtuse angles that are greater than 9000:04:22
degrees should be added here00:04:25
the unfolded angle is the angle that sat on00:04:27
twine and its size is 180 degrees00:04:32
many schoolchildren do not understand why00:04:35
exactly 180 guys the thing is that once00:04:38
they took it a long time ago00:04:40
degree measure of a complete circle or00:04:44
circles in 360 degrees, that is00:04:48
wrap around a circle of 35000:04:50
degrees there is a myth that00:04:53
Where did the number even come from?00:04:55
there is a myth that they did this because00:04:58
that the earth goes around the sun in 365 days00:05:02
turns around but it's just a myth00:05:05
in fact, it was just in those days00:05:07
popular numbers are 60 120 360 pure00:05:12
which is divisible by 60 at that time was considered00:05:14
6 tens now we are used to00:05:17
count00:05:18
in tens 10 20 30 40 we even have00:05:22
money00:05:23
100 rubles are considered tens 20000:05:26
rubles 500 rubles thousand so and then00:05:29
time there were 6 tens exactly00:05:31
for this reason00:05:33
that is, they could easily take a whole00:05:35
a circle of 400 degrees by the way00:05:37
unit of measurement is it's called00:05:39
hail but they took 360 and so it happened00:05:43
and therefore half a circle is 18000:05:46
degrees and00:05:48
the height of a quarter circle is 90 degrees00:05:51
that's why it just happened that way00:05:54
ok and now we come to the next one00:05:59
next to next topic adjacent angles00:06:02
adjacent angles from a word00:06:05
there is such a word as boundary or something else00:06:08
adjacent can be translated as neighboring00:06:11
corners, this is their country, these are neighboring00:06:16
it's like two neighbors who have something in common00:06:19
fence two neighbors to a neighbor whose common00:06:23
fence00:06:24
And00:06:26
we see what they add up to00:06:28
the turned angle is therefore the sum 18000:06:30
degrees but also God and you can00:06:33
see00:06:34
for example, not two angles, but maybe00:06:37
for example three angles which also add up00:06:40
give a straight angle00:06:42
such loonies are called adjacent only00:06:46
this picture is called adjacent00:06:47
angles this picture is not called with00:06:49
by the way, but nevertheless their sum gives00:06:51
180 degrees just because00:06:54
they add up to a sweep00:06:57
ok we've sorted out the angles now00:07:00
we move gradually do not fully feel00:07:03
vertical angles vertical angles are00:07:05
angles that are on top of each other like this00:07:07
look they turn out here is angle 1 and angle00:07:10
2 they are obtained if two straight lines between c00:07:13
cross yourself00:07:14
you know, here are 3 more 4 corners the same yes00:07:17
they also look at each other00:07:19
know it's very intuitive00:07:21
the rule is actually possible here00:07:24
imagine scissors, this is the handle and00:07:27
these blades are naturally the stronger00:07:31
You00:07:32
the more you open the handle00:07:35
the blades also open, so the angles and 1 and00:07:39
2 non-intuitive images are equal for us and00:07:43
So00:07:45
the next moment is of course the views00:07:47
triangles00:07:48
how about taxes with corners we have sharp00:07:52
angle right angle obtuse angle such00:07:55
maybe triangles00:07:57
acute-angled00:08:00
right triangle and00:08:02
obtuse triangle00:08:06
obtuse00:08:09
there's not much to say here, but here it is00:08:11
There are also separate types of triangles00:08:13
it is an isosceles triangle and00:08:16
equilateral triangle00:08:18
equilateral00:08:20
differs in many ways in that it has00:08:22
all all all sides are equal but00:08:24
isosceles this is so when only00:08:26
only two sides are equal now00:08:28
let's talk to each one separately00:08:31
let's figure it out with a triangle00:08:33
scalene triangle00:08:36
isosceles triangle its two00:08:39
sides00:08:41
are written called as hips00:08:45
or lateral sides made a reservation lateral00:08:49
sides a00:08:50
third party which is not equal to this one00:08:53
the side is called the base00:08:55
and look very interesting here00:08:58
Many people don’t understand what a foundation is.00:09:00
in fact, we consider the basis00:09:02
any side that is just below00:09:04
there is, that is, a basis in fact this00:09:05
side00:09:06
side and if you look at it00:09:09
an ordinary triangle in which00:09:11
any country is unequal00:09:14
may be the base so which is below00:09:16
for example, now the basis is00:09:19
this side00:09:20
but if you take it, turn it over00:09:22
the triangle doesn't even have to be turned over00:09:24
just see if your head is like this00:09:27
tilt it, then the left side will be00:09:32
the reason is so00:09:33
that is, in theory there can be any country00:09:36
base but at an isosceles00:09:39
triangle00:09:40
only one country is always the basis even00:09:43
if he is lying on his side00:09:45
that's kind of it, it got full as they say and00:09:48
he's sleeping, that is, these are his sides00:09:51
the bottom side is not the same00:09:55
called a basis, that is, if you00:09:57
the condition of the task says here it is00:09:58
isosceles triangle base y00:09:59
it equals 3, which means we are talking about this00:10:02
side about00:10:05
the one that is not equal to the other two00:10:08
the sides keep this in mind that00:10:11
base of an isosceles triangle00:10:13
a little differently they say that00:10:17
mathematics is definitely a science but there are inaccuracies here00:10:19
there are actually a lot of jargons of all sorts and00:10:22
so the property of isosceles00:10:23
triangle a would be equal to bc that is00:10:26
Of course, these sides are equal and00:10:29
the angles at the base are also equal00:10:32
the angles at the base are also equal and this00:10:35
can be easily understood, look firstly, well00:10:39
purely intuitively since the triangle is00:10:43
yes yes and its two sides are equal00:10:44
it turns out it is completely symmetrical then00:10:46
Yes, if you draw the axis like this00:10:47
symmetry then everything on the left is equal to that00:10:49
what's on the right is even these segments00:10:52
nor hurricanes and these corners are also equal00:10:54
this symmetry is healthy for us00:10:58
this is intuitive with you because00:10:59
We are people, we are also symmetrical00:11:01
our left and right hands are the same and00:11:03
the eyes are also the same, well, almost everyone00:11:06
that's00:11:11
but you can prove it geometrically00:11:13
look here our sides are the same00:11:17
so imagine these two in short00:11:20
triangle a bh a bh and hbc they are both00:11:24
straight if rectangular if draw00:11:27
the height is secondly they have sides00:11:31
the same is 2 in00:11:34
they have third00:11:39
the same00:11:41
common side and00:11:45
this is actually enough to00:11:48
prove that the triangles are congruent because00:11:50
that they have two rectangular ones00:11:51
triangle to rectangular00:11:53
the triangles were equal enough if00:11:55
they have 29 hypotenuses and one leg is equal00:11:58
so we'll look at it later00:12:00
equality of triangles and and later this00:12:02
You’ll understand the topic better in this video00:12:05
more knife, all geometry is good00:12:07
now the third point is height and god00:12:10
is the median and bisector00:12:13
what does this mean what does it mean that if you00:12:15
lower the height, it will be like three00:12:18
in one it will also be the median median00:12:21
it divides the opposite side in half00:12:23
there will be bisectors it divides the angle00:12:25
it's not necessary at all00:12:28
this happens in all triangles00:12:30
occurs only in isosceles00:12:31
triangle if you are in an ordinary00:12:34
triangles are just ordinary00:12:36
let's say drop the triangle here00:12:38
the height will be one segment, lower it00:12:42
the bisector divides its angle in half00:12:44
this will be the second segment and lower00:12:46
the median but in fact the same side in half00:12:50
it will be three different segments not only in00:12:53
isosceles triangle they are all00:12:54
match will also not match00:12:57
equilateral triangle because00:12:59
equilateral triangle00:13:00
partly00:13:02
isosceles so you take 200:13:04
sides are equal to at least 3 as it goes00:13:07
a bonus therefore everything that concerns00:13:08
isosceles triangle00:13:10
people's side triangle too00:13:12
copied now equilateral00:13:16
triangle first of all, all sides are equal00:13:19
because equilateral at second angles00:13:23
with him, too, everyone is equal and it’s easy00:13:25
calculate what each is equal to because00:13:28
the sum of all angles in a triangle is always00:13:30
Divide 180 by 3 to get 60 that's why00:13:33
everything will be 60 height is the median00:13:36
and a bisector with which any in00:13:39
in an isosceles triangle only one00:13:40
which is lowered to a height for example if00:13:44
this is this, this is three in one, this is this00:13:47
the height is no longer three in one00:13:50
isosceles in the mouth side all 300:13:52
these are all three heights he has, these are three00:13:57
in one00:13:58
Fine00:14:02
from further next slide see more00:14:07
there are such forms that for some reason00:14:10
rarely writes rarely shows00:14:14
actually it's very useful on the water00:14:16
by the way it is given there00:14:18
for example if you are given height if you00:14:21
given the side of a triangle then you can find00:14:24
in height and this formula is given by and and00:14:27
there is no need to remember, therefore it is a sin and00:14:29
I can't tell him of course00:14:31
very conveniently took our height works00:14:35
only equilateral triangle to00:14:36
unfortunately this is actually a formula00:14:38
is derived through trigonometry which we00:14:40
Let's go soon, I'll explain this formula to you00:14:42
we will bring her out to them in any way together with you too00:14:45
nougat these two formulas are given:00:14:47
inscribed circle radius and radius00:14:50
circumscribed circle they again00:14:52
work only for the sake of outsiders00:14:53
triangle I'll tell you one00:14:56
an interesting pattern, well, first of all00:14:58
what you need to know here look at the letter r00:15:00
small, it means we are talking about00:15:02
small circle if or large00:15:04
this means we are talking about a large radius00:15:06
circle described so that's not enough on00:15:11
really pay attention here00:15:13
if you take a triangle describe it and00:15:17
internal inscribe then the radius of the inscribed00:15:20
circles and00:15:22
circumscribed circle radius00:15:25
here I am just Nikita R small radius00:15:28
this is r this is small here they are together00:15:33
they just form this height00:15:35
can be seen from the picture00:15:38
this is a law supporter of the triangle and00:15:41
again00:15:42
Fine00:15:44
Fine00:15:45
these forms need no explanation here00:15:49
just use them, we take them out and00:15:50
we won't sit down because they give up00:15:52
many things are now medians and bisectors00:15:54
the height of the median is divided by the opposite00:15:57
side in half as I said00:15:58
the bisector bisects the angle and the height00:16:01
it is perpendicular to the opposite one00:16:03
side now in detail about each of00:16:05
we will still tell more segments for now00:16:08
that's how it is, that is, this is the median00:16:10
bisector height00:16:13
Many people have problems with height because00:16:16
that many people don’t understand what height is00:16:19
This is especially evident in problems on the area00:16:23
because height is required everywhere00:16:25
now the triangles are equal00:16:28
congruence of triangles at school00:16:30
they say that the triangles are equal and00:16:33
if when superimposed on each other then they00:16:36
match, well, in short, you are clones00:16:38
there is a coupe of each other, I copied it00:16:40
inserted00:16:41
it is natural that they will only be equal00:16:45
if each element is equal, that is00:16:47
all three corners and all three sides00:16:49
but you understand that00:16:54
in short, when they came up with signs of equality00:16:57
triangles and total 3 1 2 300:17:00
why they were invented not to00:17:03
to complicate your life on the contrary00:17:05
to make it easier for you, the thing is to00:17:08
prove that two triangles are not equal00:17:12
you need to prove that everything is everything00:17:13
the elements are equal it turns out to be enough00:17:16
several00:17:17
for example when we see her car00:17:20
What are the signs of her mind? She's roaring here00:17:23
motor00:17:24
She's going fast, she's rolling, she's got four00:17:26
wheels00:17:27
what else is she dear, especially now00:17:31
in general she has a lot of signs00:17:33
but if we even just hear the roar00:17:38
we don’t even need to see the engine00:17:40
the car itself we understand that this car00:17:42
we only need one sign for this00:17:45
understand this in the same way in equality00:17:47
we only need triangles00:17:49
several signs to prove that00:17:51
triangles are equal00:17:53
the first sign is00:17:56
through two sides and the angle between them00:18:01
and look, many people think it’s possible00:18:04
just two sides and any corner no then00:18:06
there is this corner, it’s no longer00:18:08
fits, that is, it is important that the angle is00:18:09
between these countries00:18:12
this is an important point00:18:13
Even excellent students, some people think that00:18:17
so maybe no the angle should be exactly00:18:19
between them why is it like this guys, here you go00:18:22
judge why this is a sign he is very00:18:24
logical and understandable, just imagine00:18:26
next 3 parts for the triangle00:18:29
They said there is a country, the length of the segment00:18:33
length 5 segment length 6 and00:18:36
corner00:18:37
30 degrees is acceptable and00:18:40
they say assemble a triangle from these00:18:43
you will say that I don’t have three details00:18:44
Don’t rub my other parts as strange00:18:46
to the other two corners, but the thing is that00:18:49
these details will be enough for you00:18:52
try to collect, I'll take 5 to 4500:18:56
marking die off with a protractor that is00:18:58
measured the ruler first 5 from not give00:19:00
protractor angle 30 degrees00:19:02
here I measure 30 degrees on this line 600:19:07
centimeters00:19:08
watch me automatically00:19:10
2 and 3 were built. I asked the first one myself00:19:15
triangle automatically adjusted00:19:17
we don't need to know this side of her00:19:19
builds on its own00:19:21
understand why now it's a sign00:19:24
triangle congruence is exactly like this00:19:26
because you know these two sides and the angle00:19:29
enough to make a triangle00:19:31
you can't build another one00:19:33
a triangle with these 3-dimensional at least00:19:36
the company data is still not different00:19:38
everything will turn out the same it will work out on its own00:19:41
there is for example if you first 6 postpone00:19:43
and then 500:19:44
nothing will change it will be the same00:19:47
another question inverted00:19:49
well the same goes for 2 signs00:19:53
it turns out it can be proven that two00:19:55
triangles are equal just by knowing00:19:57
side well, they have equal sides and00:20:00
two angles on this side or else00:20:03
they say two corners and a country between them00:20:05
By the way, many people think that you need to memorize00:20:07
feature numbers no guys in the solution00:20:10
problems you can say these triangles00:20:12
equal not on the first grounds not on00:20:14
the second one, that is, no one really cares00:20:16
remember the numbers of signs say00:20:19
equal in two corners and the country between them00:20:22
that is, they speak directly in words00:20:25
Guys, why is this a sign?00:20:29
exists because if you were given00:20:31
for example 3 parts this country is acceptable00:20:33
6 and 2 corners00:20:35
40 and 30 to make up from these details00:20:40
triangle you take the ruler at mark 600:20:42
centimeters00:20:43
we transport measure there are even such00:20:45
construction problem in the textbook tenase00:20:48
she measure 30 40 degrees to00:20:51
with a protractor they cross you00:20:53
automatically in this one. that is, after all00:20:55
you . was born on its own, that is00:20:57
a triangle is born00:21:00
already from these three elements and00:21:03
you can't do it any other way00:21:06
you can swap the corners, well00:21:09
the triangle will turn over but it will be00:21:10
the same triangle00:21:12
you understand well, and the third sign is00:21:14
just on three sides in three countries00:21:16
yes, if you are given three details, that is00:21:19
three sides, otherwise they cannot be collected00:21:21
it will always work out one and00:21:22
the same triangle it could be00:21:23
turned in different directions but that's all00:21:25
the same triangle00:21:27
ok now signs of parallelism oh00:21:31
In short, what are we talking about here?00:21:33
parallel lines you know that00:21:35
parallel lines are those lines00:21:37
which do not intersect in life you are00:21:38
For example, you constantly see curbs00:21:40
curbs never intersect if you00:21:42
if we hadn't crossed the road it would have ended00:21:45
ok now but how to determine how00:21:49
prove that the lines are parallel00:21:50
we can’t do them indefinitely00:21:52
We'll run out of foxes to draw.00:21:55
there are signs specially invented also00:21:58
same as with triangles00:22:00
it turns out, but if you do this00:22:03
a secant creates many angles00:22:06
if between the corners00:22:08
certain relationships arise00:22:11
some kind of equality arises here00:22:14
it turns out00:22:16
one line parallel to another00:22:19
let's get this drawing here already00:22:21
drawn street and paired rovina00:22:24
1234 numbered five six seven00:22:27
eight00:22:28
ok now00:22:31
It turns out these corners have their own names00:22:34
for example, crossed angles, what are they?00:22:38
crosswise angles turns out to be00:22:40
angles 3 and 6 for example and00:22:45
if angles if if ours are straight00:22:48
are parallel then they are equal00:22:51
why are they called that so weird00:22:54
words I never understood what they meant00:22:55
crosswise maybe because I'm not Russian00:22:57
of course, but I really didn’t understand that00:23:01
It’s very easy to do this on the cross00:23:03
look the thing is that they are real00:23:05
lie crosswise, that is, this one goes here00:23:08
looks a 6 looks the other way a00:23:10
not like knights running at each other00:23:12
remember those medieval films where00:23:15
knights run at each other and try00:23:17
whip each other with a spear, knock each other off a horse00:23:20
and and now they look like these knights they00:23:23
running criss-cross this is the first moment in00:23:26
English literature is much more00:23:28
they have a convenient name they are called z00:23:32
corners because they form the letters z00:23:35
so here is angle 3 yoga 6 if they are equal00:23:40
then it follows that straight00:23:43
parallel00:23:44
the thing is that we don't think that they00:23:47
these crossed angles are always equal but00:23:48
they are not equal dear friends, look00:23:50
I’ll give you an example, they’re not parallel00:23:52
straight and here, sorry for the cheek and here00:23:55
there are also here they are but they are not equal they00:23:57
they're different, even this one, but you can see this one00:24:01
the angle is larger yes that was more than00:24:03
You00:24:04
even the look is visible so if only00:24:08
they are equal, but here some are equal only in00:24:10
in this case the lines are parallel like this and00:24:13
it works in reverse too if it's straight00:24:16
parallel then to the coloring one you are equal to this00:24:19
works both back and forth from twig like00:24:22
I would have come well00:24:26
that is, how can you remember if the hats00:24:29
the angles of the letter z are parallel and equal if00:24:32
parallel to me unequal and vice versa too00:24:34
it works let's give an example like this00:24:36
many people get confused, in short, there is this one00:24:39
Problem in the first part in Ohio trapezoid00:24:41
given you need to find this angle up to 3000:24:44
Let's say you're asking for this one here00:24:48
one more angle there can be given00:24:49
let's say 40, this angle is asked and00:24:51
we don't answer 30 many schoolchildren00:24:54
they say 30 well, right there lying crosswise00:24:56
corners here please say here here here00:24:58
this z is drawn crosswise please00:25:01
lying angles they are lying crosswise but00:25:03
straight lines are not parallel I'm not even00:25:04
She has a trapezoid, they intersect there00:25:07
somewhere further that's why the answer is to me00:25:09
thirty naturally needs to be decided here00:25:10
differently and so angle 3 is equal to new 6 but00:25:13
there are also angles lying crosswise here, this is 4500:25:16
you also see that they form the same letter z00:25:19
she's just stretched00:25:20
I don't look at each other like that either00:25:24
here on the cross it can, that is, this is an angle of 500:25:26
corner 4 good now00:25:29
so there's a typo here there's a typo00:25:32
We're talking about one-sided ones here.00:25:35
one-sided angles00:25:39
one-sided angles are equal to 180 and here00:25:43
vice versa accordingly00:25:45
corresponding angles00:25:47
corresponding angles00:25:49
equal00:25:51
let's start with the corresponding corners guys00:25:53
in short, the corresponding angles are00:25:55
clones00:25:57
clones you see here one gang of corners00:26:01
here's another gang of angles and they are00:26:04
copies of each other, that is, here is the first00:26:06
corner copy 5 to the second corner is copy 600:26:12
angle, that is, they are each other00:26:14
correspond correspond therefore you00:26:17
accordingly, this is the easiest thing to remember00:26:19
that is, angle 1 and angle 5 for example00:26:22
corresponding angle 2 and angle 6 on the cotton wool00:26:25
the rest even 3 3 4 5 6 7 800:26:29
800:26:31
3 and 7 in general there are a whole bunch of them here not with00:26:35
there are a lot of them and some have 4 of them00:26:37
it turns out four pairs about 8 corner 400:26:41
like this00:26:42
if they are equal then the lines are parallel00:26:45
and vice versa also works now00:26:48
what are one-sided angles?00:26:50
side angles and they lie one at a time00:26:52
side of parallel lines for example00:26:55
corner 6 and corner 4 they both lie inside00:27:00
in one room they have their own ace from their00:27:03
topic here they are one-sided angles00:27:05
are called00:27:06
internal on the street side and there is more00:27:07
externally, for example, the angle is 2 and they are both about 800:27:10
on the outside they are called00:27:12
external on third-party corners but this is not00:27:14
you need to, you can do without it, so here’s 500:27:18
3 also turns out to be one-sided angles00:27:21
the sum of one-sided angles is 18000:27:22
degrees if the lines are parallel, that is00:27:26
angle angle 6 plus angle 4 equals 180 and angle00:27:31
3 plus angle 5 equals 180 that's it00:27:34
one-sided angles why 180 look at00:27:36
what a joke00:27:37
angle 6 is the same with angle 3, that is00:27:41
they are equal and now look at the angle at 300:27:45
angle 4 they are adjacent they form together00:27:49
turned angle that sat on the splits00:27:52
remember the UN 180 would be deployed that’s why00:27:56
sum of one-sided angles 18000:28:00
that is, we have a flashback, we need from00:28:03
topics adjacent angles I will try so00:28:05
link several topics so that you can00:28:07
it's easier to understand everything00:28:09
Fine00:28:12
why did I tell all this the thing is00:28:14
what is all this00:28:16
then meets in quadrangles00:28:19
for example one-sided you really are00:28:21
I evaporated the gram because it’s right here00:28:23
you can draw programs here he appears00:28:25
this is a piece of a kilogram so he can have it00:28:28
the sides are parallel and00:28:30
she has a trapezoid, she has the same thing00:28:32
the base is parallel and just imagine00:28:35
what is this side side this too00:28:36
let's say the side then00:28:38
the trapezoid obtains the sum of the lateral00:28:40
the angles are 180 just because of that00:28:43
which is not one-sided00:28:44
now further00:28:47
What kind of quadrilaterals are there?00:28:50
quadrilaterals00:28:51
Just imagine a diagram like this:00:28:54
this circle is all quadrilaterals00:28:56
which exist here separately00:28:59
separate gang stand trapeze00:29:03
trapeze next huge diaspora00:29:08
parallelograms, look, then let00:29:11
it will be like this00:29:12
parallelograms00:29:15
parallelogram00:29:18
there is a parallelogram and who had it00:29:22
direct, that is, if you took it here00:29:24
they would have taken almost no programs00:29:26
tabasarana hours nuruk we took it00:29:28
repaired and the angles are now 9000:29:30
degrees this is called pour la gran00:29:32
a rectangle that is a figure00:29:34
rectangle she lives among the rule00:29:39
grams in their diaspora00:29:42
now look if you take a rectangle00:29:45
which has right angles all and make from00:29:48
him square00:29:50
that is, make him the same00:29:52
sides, that is, we have diasporas00:29:53
rectangle00:29:55
then it turns out to be a square I in00:29:58
here I will draw00:29:59
he was a separate group until00:30:01
actually let's not rush00:30:03
the thing is that there is also a rhombus there is a rhombus and00:30:06
this is a program whose sides are equal then00:30:08
took tamed parla gram to have him00:30:10
all sides were equal and it turned out to be a rhombus00:30:14
here is rob and if you cross rob and programs00:30:20
if you cross the squares the gram rule then00:30:22
it turns out to paint a square, I’ll explain now00:30:24
look at the run it's a program00:30:27
initially but there is also cheese to make00:30:30
right angles that is, make it out of it00:30:32
there is a rectangle out of the coffin00:30:34
it turns out to be a square because it's a rhombus00:30:36
originally had the same sides and00:30:38
so it turns out to be a square00:30:41
here's a diagram like this00:30:45
it means if you study00:30:47
properties of a parallelogram then you will00:30:49
know the properties of both rhombus and rectangles00:30:52
and they are all squares because they are in one00:30:55
diaspora these are all rules gram no00:30:57
rectangle square and rhombus but00:31:00
but yes that's right and here what's here00:31:05
just some stupid, ill-mannered people00:31:08
quadrilaterals with all sides00:31:10
and the angles are different00:31:12
which humpty dumpty are now properties00:31:15
parallelogram00:31:17
what's going on here, let's go in order and00:31:20
so the properties of parla gram well, first of all00:31:23
him00:31:24
opposite sides are parallel and00:31:27
red and blue and00:31:30
so they are equal well the thing is that00:31:33
parallel lines00:31:36
parallel lines00:31:40
well, in general they are equal and yes now they are00:31:43
equal and parallel and opposite00:31:46
their angles are also equal, but this is not00:31:48
It's hard to believe but it really is visible00:31:49
that they are equal in any drawings you usually00:31:51
you can easily guess what kind of smile they have00:31:53
in parallel you will never say that00:31:54
these two angles are equal to usually with this00:31:57
you have no problems with this00:32:00
there is a problem, I'll tell you now00:32:02
here it also says that the diagonals are in00:32:04
the point of intersection is divided in half is00:32:06
the truth is that it is like this00:32:09
are divided in half but many people think that00:32:12
they also have the same diagonal but00:32:13
this is not true of course, look at me00:32:15
I’ll draw some programs for you now and you’ll see00:32:17
that their diagonals are one hundred percent different00:32:21
just about a prologue here he has 1 of 2 you00:32:25
you see that they are super different they never00:32:28
are not equal they are only equal00:32:30
at the rectangle now when it00:32:32
we'll get there okay, they still think it's a diagonal00:32:37
is a bisector00:32:38
but it seems so yes, why not00:32:40
why isn't she a bisector?00:32:41
bisector dear friends to do this00:32:43
to feel so that it is so that00:32:45
be filled with understanding of programs00:32:48
I'll show you here and pieces of programs you00:32:50
see that he has a diagonal00:32:52
not a bisector at all, well there you go00:32:53
look how big the angle is here00:32:55
how small it is, it's not a bisector00:32:57
no way, so don't say00:32:59
the problem is that the diagonal is a bisector00:33:02
now the sum of the angles adjacent to one00:33:06
side is equal to 180 degrees, that's it00:33:08
the interesting and most important thing is the amount00:33:11
angles a and b 18000:33:13
why because there are parallel00:33:16
parallel lines are a secant and so00:33:19
these are the same one-sided ones you are talking about00:33:20
which we just said remember00:33:22
they are equal because they their sum is equal00:33:24
180 because the angle is equal to this00:33:29
the corner that lies criss-crossed00:33:32
lying on the corner they together form already00:33:34
unfolded was who sat on the splits00:33:36
180 degrees like this and00:33:39
flashback zeros in00:33:41
snowy you see yes yes yes good00:33:46
she'll leave it like that00:33:49
all other adjacent angles, that is,00:33:51
these ones too, they are also one-sided and00:33:54
these too because look here00:33:56
here are parallel lines here is a secant00:33:58
you see these two corners in one room00:34:01
they sit one-sided they are one at a time00:34:03
side of parallel lines is good00:34:06
and the whole same story on the website00:34:09
trapezoids now a little more about it00:34:11
I'll tell you00:34:12
first let's analyze the rectangle first00:34:15
what about a rectangle you need to know everything00:34:18
the same as the program exactly00:34:20
same properties because00:34:21
rectangle and logran axes and more00:34:23
all its angles are equal to 90 degrees and00:34:26
its diagonals are equal, that is, it is like this00:34:28
envelope imagine an envelope right there00:34:30
the envelopes are also the same diagonal00:34:32
carried out here, this one is usually many00:34:36
lick with tongue and stick here an envelope00:34:39
and these diagonals are equal well00:34:42
that is, if you want to build00:34:44
a rectangle can be complicated00:34:46
first equal to the diagonal draw a00:34:47
then draw the rectangle yourself00:34:50
properties of the program is good, that is, here00:34:52
nothing special00:34:53
Many people still don’t understand why00:34:56
here its diagonals are equal and equal00:34:59
they are because they are halves00:35:01
diagonals are equal why because y00:35:03
Karla gram they are divided in half and here00:35:06
The diagonals themselves are equal to their half00:35:08
also equal, keep in mind that is necessary00:35:10
remember yourself about the parallelogram place00:35:12
Well, it’s just a square, that’s all00:35:15
sides are equal and plus property of raw rhombuses00:35:18
and programs and rectangle00:35:21
now the property of a rhombus is here again00:35:24
the same thing as flogged gram but still00:35:26
all sides are equal and look what00:35:29
beauty diagonal is a bisector00:35:32
lesson, but the program didn’t have that00:35:35
it was, it wasn't, it was still very short00:35:40
interesting property00:35:41
of a rhombus, the diagonals are perpendicular00:35:44
only the rhombus has no gram so covered and00:35:50
because of this, by the way, the rhombus is divided00:35:53
into four identical rectangular00:35:55
of a triangle, that is, its area can be00:35:57
first find one triangle then00:35:59
Just multiply by 4 what next?00:36:04
All00:36:08
the trail is a trapezoid in fact00:36:13
unfulfilled rule gram that is he00:36:15
tried to become a programmer but he00:36:17
It was only half done, but here it is00:36:19
his other country is not parallel to the axis00:36:21
on 2 sides parallel to the other two00:36:23
no, the boy was on his way to success00:36:27
By the way, that's all the reasons for the program00:36:30
four sides are the grounds up to can00:36:33
be that is the underside of the base but00:36:35
if you put the other side down then00:36:38
it will also be the basis of any country00:36:40
can be a reason so if desired00:36:42
when you are looking for the area of a trapezoid only00:36:45
two bases, there's an area across00:36:48
I can't find the sides00:36:50
only through the base00:36:52
good sum of angles at the side00:36:55
equal to 180 degrees, that is, the angle alt +00:36:57
angle b 18000:36:59
and here is a typo angle a plus angle b is equal00:37:03
180 degrees and angle c plus angle d is equal to00:37:07
180 degrees like this and like this00:37:12
look, many people think that for example if00:37:15
you solve his problem, yes00:37:18
trapezoid isosceles but in principle00:37:20
that's it, you have to understand that these two00:37:22
angle and these two angles are in no way interconnected00:37:24
connected that is, if here let's say 3000:37:26
you can find this one because 18000:37:29
minus 30 to 150, so the sum is also 18000:37:32
yes, but you can’t do this angle00:37:34
find they are not related he they are not even00:37:39
so that relatives understand simply00:37:40
everyone lives their own life, they don’t know00:37:44
they don’t call about each other, but suddenly there is00:37:48
isosceles trapezoid for example00:37:50
such a case00:37:51
then it becomes symmetrical00:37:53
automatically . here is the axis of symmetry00:37:55
and everything everything on the left is equal to everything that00:37:58
because right and left and right00:38:01
equals and00:38:02
many people think many people are mistaken for example00:38:06
here it is given this is 30 until it is clear what is here00:38:08
will be 18000:38:09
30 that is, it will be 150 and many people think ah00:38:13
here too then here too 18000:38:17
minus 30 will also be 150 of course not00:38:19
these 2 angles are not connected, they are connected00:38:22
I just stumbled upon something like this here00:38:25
you can't say that, they can be00:38:26
only equal or not at all00:38:28
connected they will be equal if trapezoid00:38:30
isosceles Fine00:38:32
Well, now the midline of the trapezoid00:38:35
the midline it divides the sides00:38:38
in half and it is parallel to two00:38:40
base, that is, there are three parallel00:38:42
found 3 pro arrenas direct turns out00:38:43
and it is equal to the arithmetic mean00:38:46
between the lower and upper base it is00:38:49
it’s logical because you can see even by00:38:51
The picture shows that it is a little longer00:38:53
short side and slightly shorter than the long side00:38:55
sides00:38:58
ok now00:39:00
right triangle00:39:03
the favor and the most important thing is that you should00:39:07
know the Pythagorean theorem00:39:10
the most important thing is the rectangular tower00:39:12
triangles and00:39:14
she is very powerful because if you00:39:18
you know two sides example five and three then00:39:21
you can easily by 4 you just need00:39:23
load all the numbers into this formula here00:39:27
here we don't know in and can this this00:39:31
the best solution would be to solve this equation00:39:34
spend an hour solving equations00:39:36
right triangle00:39:38
sides that are adjacent to a right angle00:39:42
called kittens, kittens are rolling in the country00:39:45
which is opposite the right angle00:39:48
hypotenuse when asking schoolchildren00:39:50
they don't say what the hypotenuse is00:39:52
the long side and even in these00:39:56
for example a triangle is not right angled00:40:00
in triangles they call the longest00:40:02
side of the hypotenuse friends this is not true00:40:04
don't say this without culture00:40:08
this is the same thing that says it rings00:40:11
don't say that wrong00:40:14
this can't be a hypotenuse00:40:16
hypotenuse because the hypotenuse is00:40:18
by definition this is the country that lies00:40:20
opposite the right angle, that is, the hypotenuse00:40:23
she was born in a right triangle00:40:26
can't be in another triangle ok00:40:28
well what does this mean for example in00:40:31
definition of sines there Katya is divided into00:40:34
where to pull it to know what sine you are00:40:36
can only be found in rectangular00:40:37
triangles and then some start00:40:38
find them in non-rectangular ones, which is00:40:41
impossible about them now we will00:40:43
talk about sine and cosine and currents00:40:46
Now there is a theorem which is still in00:40:49
seventh grade are studied but in fact00:40:51
it's related to trigonometry now00:40:53
I’ll explain, and so it rolls opposite the angle of 3000:40:56
degrees turns out if you build00:40:58
right triangle with angle 3000:41:00
degrees00:41:01
it's so well built here that00:41:05
so it coincided that the one who rolls is the one who lies00:41:08
opposite him is exactly half the size00:41:11
hypotenuse that is there is 10 here then00:41:14
here it will be 10 in half, it's simple00:41:16
unique triangle we are a star but00:41:19
it happens, well, just wow for others00:41:22
only he has a triangle with stones00:41:25
what are the other sides equal to you already00:41:28
will help you find sine and cosine and or 3-700:41:30
fagor now let me tell you about00:41:33
Pythagorean theorem00:41:36
herbal proteins yes00:41:38
So look, you're in ninth grade00:41:42
study but until the ninth grade you00:41:43
study sine and cosine and only acute ones00:41:47
angles sine and cosine and and dance and00:41:51
sharp corners, okay, at least a day00:41:54
dance and but about for I don’t want about at all00:41:57
tell him because he's just00:41:58
it never takes up space on the slide00:42:01
many things are used00:42:02
sine and cosine and acute angles then you00:42:06
a little later someone in high school00:42:08
here you have time to pass the ninth grade00:42:11
in general, then you study obtuse angles00:42:14
sine and cosine at obtuse angles is different00:42:17
topic there you need to know other things now00:42:20
in the game you only need to know the acute angles00:42:24
body of space of sharp angles00:42:27
now what is sine and cosine acute00:42:29
corners00:42:30
these are the uniforms they throw00:42:33
navaga are given if something is right for me00:42:36
in words it's written there, it's simply written there00:42:38
since through letters and so is sine00:42:42
divide the opposite leg into00:42:43
hypotenuse adjacent cosine what does it mean00:42:46
opposite00:42:47
lies opposite, that is, here you have00:42:50
angle a and opposite00:42:53
like the opposite, that is, imagine that00:42:56
you have these corners, here I will become the place of the corner00:42:59
So I look across the street and there are kittens lying00:43:02
bc this is how you need to introduce yourself video00:43:05
this angle is good and divide by the hypotenuse00:43:09
let us show everything with an example so that00:43:11
it was more convenient for you, this 3 let it be00:43:13
this is four this is five sine of angle a00:43:17
it will be clear00:43:19
divide the opposite leg into00:43:21
hypotenuse that is 3 divided by 5 okay00:43:23
course we now have a corner and also from below00:43:27
five will be the hypotenuse but now00:43:29
take the adjacent leg that is towards you00:43:31
adjacent which is next to your corner00:43:35
this is this leg of the father this is 4 all now00:43:42
look at the most interesting thing, that is, here00:43:43
rolls adjacent yes now the most00:43:45
interesting thing is the sine and cosine of angle b00:43:51
the sine of angle b and the cosine of angle b are here00:43:55
for those who did not understand trigonometry00:43:58
inspiration comes when you show00:44:01
look at angle b00:44:04
the thing is that these formulas are00:44:08
relate to the angle that is, you are relative00:44:10
always look at the angle if you have chosen00:44:12
angle b carry it sine and cosine need00:44:15
look00:44:16
from his point of view you need to look not from00:44:20
this side and this side because00:44:23
that many write that the sine is also 35 not00:44:25
so it's not true it can't be either00:44:27
because the angle is different for him already00:44:29
the other and spends will be opposite here00:44:32
and this leg will be for him00:44:33
opposite here you can with this00:44:35
hand will be for him kittens 400:44:38
opposite which used to be for00:44:40
it's adjacent, you understand everything changes00:44:43
everything is relative as Einstein said00:44:47
in general you need to look from the point of view00:44:49
your corner00:44:51
for angle b now the adjacent will be rolls00:44:55
3 although earlier he was against lying it00:44:58
normal story people change to00:45:01
in general, this is how it would be correctly understood00:45:04
you need to look from the angle point of view then00:45:07
it will be right now it will be on its own00:45:09
we'll figure it out, okay so be it dance00:45:12
We won’t hurt him either00:45:19
tak who ne s bk dance b00:45:22
Well, in general, you don’t need it, it’s like us00:45:26
speedrun so you probably don't need it00:45:29
because it is not used when and so00:45:31
tangent is the opposite side00:45:32
overlord on adjacent that is 300:45:35
divide by 4 here 3 divide by 4 y00:45:39
who is not in demand, on the contrary, all four00:45:41
third00:45:43
now you can dance from the point00:45:46
view of this angle for him00:45:48
opposite 4 a adjacent 3 uk a00:45:51
dance but on the contrary, that is, after all, to dance00:45:54
they just repeat you tell him00:45:56
those over there are burning, why else would anyone need them?00:45:59
needed00:46:00
remove it and we won’t fly there because00:46:03
his tasks are no longer in the game00:46:07
life hack how to remember these formulas00:46:10
after all, every time you leaf through these00:46:12
reference materials bully but it's easy00:46:15
it won’t go so well that it’s clear that they are given to you00:46:18
if you can remember anything, but you need these00:46:20
It’s good to know how to use formulas00:46:23
to do this they must be in your blood00:46:25
for them to be in the blood they need to be somehow00:46:28
associated themselves so there is one00:46:31
such a simple memory hack00:46:33
sine and cosine are a lifehack and equal to a00:46:35
that is, if here it is here with Viti00:46:37
it turns out and equals oh and here oh here he00:46:41
here it turns out levels like this and oh oh oh00:46:44
this is how the second row can remember00:46:47
look00:46:48
cosine is the second way to remember00:46:51
cosine we are so lucky as Russians that00:46:55
cosine is consonant with the word touch00:47:01
concerns00:47:03
concerns00:47:05
goats and that's why we take the kittens that00:47:11
concerns us, that is, adjacent, here we are00:47:15
you see we take 4 4 for angle a4 this00:47:17
the adjacent leg touches it therefore00:47:20
we take the adjacent leg, that’s clear00:47:22
it looks like you can come up with a story using sine00:47:25
sine looks like words looks with minus with00:47:30
and looks00:47:32
looks we take the one that rolls towards00:47:35
so look here00:47:37
the opposite side on which we are looking00:47:40
this is how you can remember using00:47:43
associations are now table value00:47:46
sine cosine dance to dance without a cat00:47:49
no connection and not even riding in an agra here00:47:51
only dance because it is not needed and00:47:54
so this is time for this in the table00:47:57
Previously, schoolchildren memorized but you00:47:59
lucky this year this table gives00:48:02
reference material and it's great00:48:04
because some people have problems with this00:48:06
table but remember it is easy00:48:08
look really you only need00:48:10
this is part of this table even this one00:48:13
part is not necessary, but this is not00:48:14
which ones to remember and00:48:17
It’s clear that this is given to you, but that’s all00:48:19
anyway, I'll show you this life hack so that you00:48:21
if they could remember something so as not to00:48:24
I know on tests for example at school00:48:26
notice they have the denominator everywhere00:48:29
deuces, that is, this is how you can write 200:48:32
2 2 goes further in increasing order00:48:36
123 is a sine and so goes cosine00:48:40
vice versa00:48:41
321 below also deuces everywhere00:48:44
22200:48:46
And00:48:47
in the numerator everything is under the root, that is, this00:48:50
under the root this is under the root this is under the root00:48:52
it's under the root it's under the root from under00:48:54
root like this, do everything that is00:48:55
just like that, fill out the whole two deuces00:48:57
one two three three two one and roots everywhere00:48:59
it's just a joke here that the root00:49:02
of units this is the same unit this is the same and00:49:05
that's why this root is not here00:49:09
but nevertheless the life hack works00:49:11
use if suddenly on a test00:49:13
forgot everything else this actually is00:49:17
the table can be expanded for you now00:49:20
I'll show you this one more00:49:22
showing off the table here it is00:49:26
angles appeared 120 135 150 here they were00:49:29
only these and now 2035 will appear00:49:32
150 in some tasks you won’t please00:49:35
would like to meet and know these angles00:49:37
you also need it in general, I told you sine00:49:40
you will learn to study obtuse angles in general00:49:43
only when you study the tags. circle00:49:45
it will be closer to the tenth grade00:49:49
there aren't any of these stupid people in the game yet00:49:52
corners but if you still suddenly00:49:55
will be needed I decided it you in detail00:49:57
insert the table here if anything and accept it00:49:59
tell you a little bit, look at00:50:00
first here you can understand that00:50:03
these are the meanings they are00:50:06
symmetrical, that is, here too00:50:08
same meaning here same00:50:09
the values here are also the same00:50:10
this is the second time the sine has a minus00:50:14
appears and cosine appears00:50:16
the dance is already appearing why are you doing this00:50:18
you'll find out when you go through the topics00:50:19
trigonometric circle00:50:21
ok, well, you can pin it for yourself00:50:24
clone the table what type if what are you00:50:26
I know all 135 to 1 yes in general it is possible00:50:29
just to put it away somewhere for yourself00:50:32
so that it was so that it was00:50:36
ok good00:50:39
there's one more thing I didn't tell you00:50:41
usually look at this slide00:50:44
many schoolchildren ask why00:50:46
will be equal to the sine of angle c and what will it be00:50:49
equal and he can't be equal to anything we00:50:51
we don't know how to count it because we00:50:53
will decrease only by the sine of acute angles00:50:55
try calculating the sine of angle c00:50:57
That00:50:58
here the opposite leg must be divided00:51:01
to the hypotenuse of the opposite side is not00:51:03
exists for him opposite lies00:51:04
hypotenuse that is, for him there is no00:51:06
opposite side respectively00:51:08
how stupid it is to count, well, you can00:51:11
Of course, get smart and try00:51:13
calculate the sine of an angle of 90 degrees if00:51:15
Let's say you build something like this00:51:18
a triangle with this angle00:51:19
almost 90 yes00:51:21
this is almost 90, it’s 89, just imagine00:51:26
that the triangle is incredibly high00:51:28
you are so incredibly tall00:51:31
kittens are making these kittens super small00:51:34
do and raise Egypt to the top00:51:36
it will be almost 90 degrees00:51:38
agree, that is, 90 is formed here00:51:41
here 90 and here almost 90 like 8 out of 9 900:51:47
if you try what is the sine of this angle00:51:49
then it will be the opposite leg00:51:50
divide by the hypotenuse about to roll here00:51:52
hypotenuse but survive what they are00:51:54
almost identical when dividing such00:51:56
the numbers will be almost one and therefore00:51:59
sine is 90 units and by the way, if00:52:02
who was interested in reading in 90 units00:52:05
and this is how you can understand yourself for now00:52:07
but in more detail you will understand it all00:52:10
when will you treat trigonometry00:52:13
the circumference is still too early for you00:52:17
ok, avoid obtuse angles00:52:21
sine and cosine of obtuse angles again00:52:24
no tax needed but just in case00:52:26
I decided to show you that it’s possible again00:52:30
versions for 150 for these it’s as simple as that00:52:33
put it in this formula00:52:35
it turns out that sine is 180 minus00:52:38
150 that is, the name in the 30s is like you00:52:42
you know the strength from 30 according to the table it will be 100:52:45
2 that is, if suddenly you see somewhere00:52:47
Paula, don't be upset if anything happens00:52:49
the island is being transformed, well, you can00:52:54
In general, find the sine of these angles00:52:56
they exist now trigonometric00:52:59
identities00:53:00
the most important thing is that you are not given any tax00:53:02
you need to know this equation00:53:04
the rest is extra I just for00:53:07
I inserted pictures of a cold, look here00:53:10
this equation helps a lot sometimes00:53:13
for example you need to find the sine and you00:53:15
given the cosine you simply substitute in00:53:17
no one can solve this equation yet00:53:19
I don’t understand, for example, I don’t understand either00:53:21
school why is this what this is all about00:53:23
what is this what does this mean this00:53:26
means that for example you take the sine of a00:53:29
and square it, that is, you00:53:32
multiply this number by itself00:53:34
it means this is how they just wrote it down00:53:39
just not because mathematicians are lazy00:53:41
they don't want to put another bracket00:53:43
we understand this is the same organism, so here we are00:53:47
speak Russian hello do00:53:50
abbreviate here is also an abbreviation00:53:52
also an organism00:53:55
there are many such abbreviations in mathematics00:53:58
for example a multiplied by b you write00:54:00
like a.b. three whole 1 3rd floor on the very00:54:03
actually 3 plus 1 3 is just a plus sign00:54:06
invisible and00:54:08
the math is actually very good00:54:11
sometimes they are accurate in moments like this00:54:14
a lot of assumptions and so on00:54:17
well in general it is clear that you00:54:22
understanding is used a little bit00:54:23
mathematics, I'll tell you where this comes from00:54:25
the formula is actually very00:54:27
very very much like the theorem00:54:30
Pythagoras especially if you are here00:54:32
put a square because 1 is in a square00:54:34
units are the same look00:54:37
something in a square something in a square00:54:39
equals something squared but it's straight00:54:41
departure we lose fagora, just replace it00:54:43
on kittens and on the hypotenuse and it turns out00:54:46
on to the paint we get lost in the mountains00:54:49
this is where the roots grow from, that is, here you are00:54:51
triangle00:54:52
rectangle a b c paint tiryns00:54:56
mountain a b squared plus b squared00:54:58
equals c squared divided and now that's it00:55:03
look at the hypotenuse of the nation squared00:55:05
what kind of ford focus is it?00:55:07
watch your fingers carefully here00:55:10
we divide nations into a square, you can do this00:55:13
if anything yes in equations in algebra here00:55:15
if your equation here is 10 you00:55:18
you can take it with you and share it raw00:55:19
at 5 nothing bad will happen00:55:21
just 2 equals 2 until the level has changed00:55:24
she stayed true it was 10 equal to 1000:55:26
2 became equal to two because you are with00:55:29
both sides of the equation did the same thing and00:55:31
same here we are doing the same thing00:55:33
it's legal and00:55:36
would like to divide the nation what is it but00:55:38
let's take this angle ab. This00:55:41
the opposite leg to the father is the hypotenuse00:55:44
it turns out it's a sine and then it's squares00:55:46
means by sine squared sine in00:55:48
squared c is what bc divided by00:55:52
hypotenuse and also cosine is cosine in00:55:56
square c and this is what this unit is you one00:56:00
and the same number is divisible by itself00:56:01
unit that's what happened00:56:05
equation other equations other00:56:07
identities are obtained from the same00:56:10
actually00:56:11
For example00:56:12
this equation is obtained if everything00:56:15
divide by cosine squared but00:56:16
relax relax again00:56:19
relax because it doesn't really matter to you00:56:21
need to00:56:23
you can be without it, this equation00:56:26
it turns out if you divide it all by00:56:28
sine squared each term here I am for you00:56:31
I haven’t told you one formula yet00:56:33
this tangent is equal to00:56:35
dance a equals sine a divided by00:56:39
cosine and where does it actually come from?00:56:41
in fact it turns out from the definition00:56:43
tangent00:56:44
the adjacent leg is his own00:56:46
opposite leg to divide by00:56:49
adjacent leg formula dance a00:56:51
initially here if from above and below00:56:53
divide by the hypotenuse00:56:54
then the formula for sine a is obtained from above00:56:57
from below we get the cosine formulas00:57:00
turns out that's where this comes from00:57:02
formula is good00:57:06
it just symbolizes that when00:57:09
I'm just kidding here, of course, what if?00:57:12
substitute all definitions of dance and to00:57:14
dance this you will see that she00:57:16
the formula is really correct, that is00:57:19
it will remain the same there adjacent leg00:57:22
divide by opposite and when00:57:23
It’s not without reason that it’s the other way around, that is,00:57:25
multiply by give by and here they are00:57:27
are reduced and it turns out to be one00:57:29
and the whole story, that is, the equation is very00:57:32
logical00:57:33
ok now the sine theorem00:57:36
brilliant tower, I’ll give you a life hack right away00:57:39
if tax tasks and you see in00:57:43
condition of the problem formulation of the type radius00:57:47
circumscribed circle is equal to something or00:57:49
if you ask the radius of the described00:57:51
circle00:57:52
that means in 99 percent of cases this is exactly00:57:57
theorem of sines because there is00:57:59
the radius of the circumscribed circle is00:58:03
look if the letter r is big I tell you00:58:05
I already said yes, this is the radius of the described00:58:08
circle00:58:09
if the radius is small then this is the radius00:58:13
inscribed circle00:58:15
meaning that it only finds00:58:18
circumscribed circle radius00:58:19
okay, what's the fun in this theorem?00:58:23
sinuses she is very cool especially when needed00:58:27
find the radius of the circumcircle00:58:29
to use it you need to understand00:58:31
Many people don’t understand the logic of this drawing00:58:34
look here the joke is that there is00:58:36
vertex a and the sides that lie00:58:39
on the contrary you are called the same00:58:41
letter only small letter see00:58:43
side a and angle a are opposite each other00:58:46
lie corner b country b corner c country tion00:58:51
small and00:58:53
what we see here we see here00:58:56
the fact that it's just sides and corners and00:58:58
which dubna lie opposite each other00:59:01
the country is divided and the sine of the angle between them00:59:03
the sine of the opposite angle is equal to everything like this00:59:07
Of course you need all this in practice00:59:09
show what to understand, well, theoretically00:59:11
can be explained this way it's all a lot00:59:13
you will understand if you solve problems00:59:15
with and losses00:59:16
ok and the next problem theorem is00:59:20
cosine and the cosine theorem00:59:23
look, remember I told you that sine00:59:25
and cosine and these are the definition00:59:28
these definitions don’t work only in00:59:30
right triangles after all00:59:32
because leg and hypotenuse are written00:59:34
they are only available in rectangular ones00:59:36
triangles so here is the theorem of sines and00:59:39
cosines works in all triangles00:59:42
and in such and all kinds in any in this their00:59:46
power that is, they are powerful weapons00:59:48
against any triangle like this00:59:50
against right triangle00:59:52
there is a Pythagorean theorem00:59:54
comprehensive there powerful and so on00:59:56
the same powerful and strong theorem00:59:59
exists for other triangles beyond01:00:02
for all other non-rectangular ones, see01:00:05
not even very similar to the theorem01:00:06
Pythagoras appeared and01:00:08
well, for example, let’s say you know01:00:11
two sides01:00:12
five and six and you need to find out somehow01:00:16
third party01:00:17
if it were a rectangular pipe either01:00:20
you just did the Pythagorean theorem and01:00:22
here what to do here it turns out for01:00:24
to find this side you need to know01:00:26
another angle that lies between these01:00:28
countries and01:00:30
This is the first formula that helps01:00:32
you need to find this side01:00:35
let's say if you know this side angle01:00:37
30 degrees then you get it01:00:39
let's look for and squared is equal to01:00:41
five squared that is b squared01:00:44
plus z squared, that is, the other two01:00:45
sides is 25 plus 36 minus 2 times01:00:50
multiply by the same sides 5 by 601:00:53
for cosine 30 oh let's make it 6801:00:56
It’s more convenient for us to calculate the cosine of 60 simply by01:00:58
2 1 2 2 cancels out 2536 is01:01:04
506101:01:06
minus 5 by 6 to 30 that is, a squared01:01:10
equals01:01:11
31 that is, a squared is equal to 3101:01:15
it turns out a is equal to the root of 31 here we are01:01:17
found it how convenient it is, that is, you can01:01:20
find any side01:01:22
triangle if you know two01:01:24
the other sides and the angles between them are very01:01:26
cool very cool good for you Volga01:01:29
this formula will be given but not like this01:01:31
here in three types yes please01:01:34
using you will be given only one01:01:35
line this line01:01:38
but it’s okay, she also uses logic01:01:41
can be understood01:01:43
notice here it's like this01:01:45
pattern here letter a here01:01:48
side here countries and here corner a01:01:50
why because the corner that lies01:01:53
between these two countries01:01:54
tell us what will be opposite to ours01:01:56
our desired side01:01:59
it's logical01:02:00
Here01:02:02
ok, in general, we need practice now01:02:07
similar triangles similar01:02:10
triangles are a very cool theme01:02:12
very vital and because look01:02:14
what are similar triangles? This is 301:02:17
walk around the corners01:02:18
the sides are respectively equal01:02:21
accordingly, proportionally well, here you go01:02:23
let's draw, look, they have angles too01:02:25
are the same and all here is angle and angle 101:02:28
the same angle b the gasoline angle is the same01:02:30
or angle c angle c 1 is also the same01:02:32
now look at the side that01:02:34
means that the sides are proportional01:02:36
means they differ by the same amount01:02:38
number of times that is if there are three01:02:41
here 6 times two and others01:02:45
the sides also need to be multiplied01:02:49
71401:02:51
56 for 5 10 and here01:02:54
that is, these numbers are the same01:02:59
this means that the sides are proportional01:03:01
if you write it down mathematically it's01:03:03
it will look like this, well, this is equality01:03:04
corners and here we just take the sides01:03:06
large triangle with detailed sides01:03:08
small triangle and always01:03:10
getting with one also works out everywhere 201:03:13
for example 6 divided by 3 is two01:03:18
ten divided by 5 is 25 everywhere 2 here01:03:22
if this holds true then the triangles01:03:25
similar triangles are similar and so01:03:28
just like in life01:03:30
daughter like her mother if mother01:03:34
intelligently well-read and brought up then and01:03:39
the daughter will be intelligent, well-mannered and01:03:42
so on, that is, they are each other01:03:46
like they're the same it's the same01:03:48
the triangle is just a different size01:03:51
ok now01:03:53
and how from the equality of triangles here01:03:56
there are also signs of similarity, that is, so that01:03:58
understand that triangles are similar to each other01:04:01
friend that to understand that he is a daughter and01:04:04
that this is a daughter, that this mother is this mother01:04:09
this particular daughter01:04:11
you just need to take a look01:04:14
I don’t know their eyes, that is, I don’t01:04:16
definitely there you can even just01:04:18
last names to understand that they are relatives yes01:04:21
that is, there are different signs01:04:23
similarities among people01:04:26
qualities of the back middle name are usually01:04:28
surname appearance we don’t know voices and so on01:04:32
further if we just see even the last name01:04:33
we definitely say that they mean daughter and01:04:37
mom and here the same thing there are several01:04:40
signs of similarity the first sign is the most01:04:43
popular and probably the most needed01:04:46
because01:04:48
95 99 percent of problems are solved precisely01:04:51
through the first sign of similarity01:04:53
triangles01:04:54
it's because it's very simple, look around01:04:56
if two angles of a triangle are equal to two01:05:00
angles of another triangle then they01:05:01
similar01:05:02
that is, if the angles are equal then not similar01:05:05
it's logical when you stretch01:05:08
a triangle whose angles do not change and01:05:10
you just need to be equal that's what01:05:12
Means01:05:14
triangles are similar if they have equal01:05:16
corners but look what's funny if you have01:05:19
two angles are already equal, even 3 is not necessary01:05:22
proving two angles is enough why01:05:25
so well, take this angle 3001:05:27
admits 40 then also for thirty forty01:05:30
the third corner he has no choice aka01:05:34
is automatically calculated, that is, it01:05:36
will be 180 minus 30 -40 this is 110 video01:05:40
automatically calculated here too01:05:42
will be 110 because the other two angles are already01:05:45
known he has no choice he has it all x01:05:47
you are like 7 rue here01:05:52
so that all three corners are even01:05:54
enough triangles that it was equal01:05:56
two corners01:05:57
this is such a wonderful sign of similarity01:06:01
why does it occur so often because01:06:03
that the equality of angles solve its problems01:06:06
it’s not difficult, one year after another01:06:08
seems equal it's easy to see that's why01:06:11
they are so popular now the second01:06:15
the third sign is not so popular but01:06:17
I still have a few words here01:06:18
which needs to be said, look many01:06:20
think that it is necessary to prove that the parties01:06:23
are equal to prove that the triangles01:06:24
similar no need to prove that they are01:06:26
on the contrary, they are unequal that they are different and01:06:29
the same number of times, that is, if01:06:31
let's say you01:06:33
calculated a would be divided by b by one b101:06:37
so you got it, let’s say 3 in 301:06:40
times more then take other sides01:06:42
that is, you take other sides01:06:46
let's say father01:06:48
father and one center also got 3 all01:06:53
now you have proven that the parties01:06:55
they are proportional and differ in01:06:57
the same number of times that's what it means01:06:59
a sign of similarity on two sides and an angle01:07:02
between them all that remains is to prove that01:07:04
they have the same ones, let's say here 303001:07:09
It’s clear that they need to be punished because they are relatives01:07:11
need to prove that they are different in01:07:14
same number of times01:07:15
Well, the third sign is super rare and01:07:18
I don’t think I’ve ever seen God at all and01:07:21
really, you just need to prove something there01:07:24
that all three sides are proportional01:07:26
remember the first ones but sometimes if you want01:07:29
straight five high mark then you need it01:07:31
remember also the second I for you I to me01:07:35
it seems unlikely that you are talking about this01:07:36
will tell but I'll tell you, I'm in general01:07:40
went through all all all all all tasks on01:07:43
geometry of God and I realized what is there01:07:45
only 4 types01:07:47
structures with similar triangles01:07:50
that is, there are various tasks there01:07:53
somewhere you need something one night somewhere01:07:55
different but the designs are the same everywhere01:07:57
there are only 4 you can do on them01:08:01
take a close look and01:08:03
I'll quickly go over where they are01:08:05
similar triangles are given here as01:08:08
el parallel now parallel ditches behind01:08:12
this turns out to be one-sided here01:08:14
angles and, accordingly, these are the angles01:08:16
correspondingly, this angle is common to them01:08:19
and it turns out that these two triangles01:08:20
just like mother and daughter in response and mother01:08:23
and here is the little daughter who01:08:26
turns out mom is pregnant daughter video01:08:29
daughter inside mom01:08:30
ok I have nothing to do with it01:08:33
the most popular design01:08:35
simple popular01:08:37
everything's okay now01:08:40
you are an old design with a trapezoid01:08:43
similar triangles appear here01:08:45
here is a small similar triangle and01:08:46
waiting a little longer01:08:50
Yes, of course, maybe a little more01:08:53
slightly different labor and will meet but in01:08:56
basically this is the most popular01:08:58
now look here it is01:09:03
corner and here there is a corner they cross01:09:05
lying therefore they are equal and here the angles01:09:07
vertical, after all, it’s at two corners01:09:09
these triangles are similar but important01:09:11
moment because they are relative to each other01:09:13
rotated 180 degrees, that is01:09:15
here if they still have the same angle01:09:18
they're spinning around this corner01:09:21
there is right here on the left side01:09:23
here and here too the sides itself01:09:25
the wall on the right is narrower01:09:26
that is, roughly speaking, if there are 10 a01:09:30
here 20 to that and here for example01:09:38
1215 not here 18 then tell me for example01:09:43
what is this side equal to?01:09:44
many schoolchildren say, right there at two01:09:46
times for parts of 20 10 before means here01:09:48
probably 9 to 18 9 but no that’s not the case01:09:52
that01:09:54
this country is like this side01:09:57
you understand because you see country 1201:10:00
it lies between these equal angles01:10:02
between two strips with one strip and01:10:04
here this country is also between these corners01:10:06
lies, that is, you need to navigate from01:10:09
corners01:10:10
it turns out rn and or you differ in two01:10:13
times means here there will be 6 they are 9 but01:10:17
here Armenian will be 9 clear that is01:10:19
they are rotated relative to each other01:10:20
like this01:10:21
they look like he has a skewer, I don't01:10:24
Could it be that it's time to spin like chickens01:10:26
grill01:10:27
now 4 example here is also similar to this01:10:31
they have a common angle, but these are their angles01:10:35
they can be proven equal through adjacent01:10:37
angles and through the properties of the inscribed01:10:40
do not describe the quadrilateral in detail01:10:42
I will if you want to understand a lot01:10:44
here it takes us a long time to explain everything01:10:47
there is this quadrilateral01:10:48
triangle this triangle similar01:10:50
they are also near the usa as if on01:10:52
the skewer rotates about 18001:10:54
degrees like this from each other, that is, one01:10:57
then to the other like this01:10:58
here in general but there are triplets here and a granddaughter01:11:02
here mom is not waiting for a huge grandmother here01:11:06
This is how I have been for three years01:11:08
the triangle is similar, look at it not01:11:10
there are photos here first of all they are01:11:11
rectangular is already one corner of all01:11:13
there is one in common and one more to go01:11:16
corner to prove everything here with mom and01:11:19
grandmothers this angle is the same general and01:11:23
daughters, this corner is the same as this one01:11:25
this can happen if everything is written down01:11:28
side01:11:29
this is a lot of task probably in the sum of the task01:11:33
Find 30-40 designs you can find01:11:36
God has a lot to decide, decide and01:11:39
you'll see01:11:40
now there is one more important thing about01:11:43
similar triangles are area and01:11:45
see perimeter of similar triangles01:11:47
provide and when01:11:49
the triangle increases, that is, it was01:11:51
small yes it was small became big01:11:53
that is, here is mother and daughter daughter’s mother01:11:56
it turns out all the elements are increasing01:11:59
but of course, for example, mom and01:12:01
foot size larger and dress size01:12:03
bigger everything all all sizes she's bigger01:12:05
And01:12:06
The area is also larger, but look at what01:12:09
funny if all linear dimensions are this01:12:12
what can you measure with a ruler?01:12:14
increase01:12:16
as many times as many times01:12:18
increased all sides example here01:12:20
we have doubled in size in all directions01:12:22
on and the perimeter will increase by 2 times and01:12:24
the height will increase 2 times and the bases01:12:26
will increase by 2 times everything, everything, everything and01:12:28
medians and bisectors - all this is possible01:12:30
measure with a ruler so all are linear01:12:32
sizes double akkas and01:12:35
area is not its linear dimension01:12:37
it is impossible to some extent01:12:39
it's increasing, it won't turn out that way01:12:41
increases by 2 times squared, that is01:12:45
here is the similarity coefficient and the area01:12:48
increases in similarity coefficient in01:12:49
squared times that is, if you have01:12:52
the triangle has increased 3 times, which means01:12:54
the cake will vary by 3 squared01:12:56
if that's why it's very logical because01:12:58
that area is 1 2 times01:13:00
base to height so you have a base01:13:04
2 increased because similar01:13:06
triangles and height increased x201:13:08
it turns out 2 by 24 that's where the square comes from01:13:11
things like this are good01:13:14
for this situation there is a task in the first01:13:18
parts even y let's say the area is given01:13:21
area of the triangle up to let's say 80 and01:13:23
given that this is this is the middle line01:13:25
m.s. and they say what the area is equal to01:13:27
this little triangle01:13:29
it is naturally twice as big as less01:13:30
because the middle line is twice01:13:32
less than the grounds and they say well here01:13:34
you have to divide 80 by 2 it will be 40 no it is01:13:37
that's not how it should be 80 divided by four01:13:39
20 because the area is 4 times less than v201:13:42
squared times less, so here it’s even possible01:13:45
if we try together four01:13:46
similar triangles like this01:13:47
look 1 2 3 4 slowly what are you doing 401:13:51
the triangle becomes theirs01:13:53
now a circle01:13:56
circle01:13:58
you need to find out the circumference first01:14:02
such inscribed central angle inscribed01:14:04
an angle is an angle whose vertex lies on01:14:06
center on the circle, that is, on the line01:14:08
this one, many people think it’s like this01:14:10
this will also be an inscribed angle no this01:14:12
it will have an uninscribed angle. must01:14:15
lie straight on the line01:14:16
circles here01:14:19
good and it turns out it is equal to half01:14:24
degree measure clade degree measure arc01:14:28
like this01:14:29
half a degree measure of arc, that is01:14:32
if it's 100 degrees here, then01:14:35
the inscribed angle of 50 will be 10001:14:39
in half01:14:40
5001:14:42
the central angle is01:14:44
directly equal to the arc, that is, here 100 then01:14:47
there is here 100 good street which01:14:51
top yo right in the center now01:14:53
the next moment of the property of the inscribed01:14:56
angle01:14:57
if this one is central and01:14:59
make the inscribed angle so that they are on01:15:01
one corner rested on one corner01:15:04
relied01:15:05
it turns out that he will not be a common arc01:15:10
that is, they rely on one common arc01:15:13
and then look what's through the central01:15:17
the angle can be found inscribed because but01:15:21
it’s clear that because of the central01:15:22
angle arc will also be equal to 100 inscribed01:15:26
then it will be 100 in half, that is01:15:28
you see it's like a relationship between01:15:29
people the central corner is as if01:15:31
the guy squeezed it in as if it were a girl01:15:34
and the arc between them is this, yes it is theirs01:15:36
connects this relationship between them if others01:15:39
they will be different, that is, if they are nothing01:15:41
does not connect then accordingly they01:15:43
are not connected in any way, that is, if it happens01:15:46
so this is the central angle to the author01:15:48
inscribed here they are not connected then01:15:50
you see they must have the same ones01:15:52
the arches have been written off, he has such an aortic arch01:15:55
central such inscribed other01:15:57
arc they are not connected so good01:16:01
now here01:16:03
see here the case when both are included01:16:06
angle01:16:07
rests on the same arc01:16:10
It’s clear that then they won’t be equal01:16:12
because but if here here xd arc01:16:15
then here x in half and here too x01:16:17
you see they are equal in half01:16:19
what does it have to do with sticking in as much as you want?01:16:21
inscribed angles they will all be equal01:16:23
because everyone rests on one arc01:16:26
ok now01:16:28
look01:16:31
there is 1 class on the theorem which is often01:16:33
are generally used quite often01:16:35
In general, one of the most popular tasks in01:16:36
circle01:16:38
angle theorem goodbye to diameter01:16:40
circle angle operations diameter01:16:42
circle of a straight line that is, if you01:16:43
take a circle and draw the diameter01:16:45
you won’t draw an inscribed angle like this01:16:46
he will be straight why watch this01:16:49
proves two counts here you have01:16:51
let's say parity 6 understand let's say if01:16:55
120 degree angle if you're on it01:16:57
enter the angle it will be 60 so you have01:17:01
whatever, now take the central angle01:17:04
do a 180, that is, do it so that01:17:06
st sat down on the splits, here he will be01:17:08
in detail there is in fact it will be01:17:10
diameter then the inscribed angle will be what01:17:12
equal to 180 in half is 90 that's why01:17:16
it turns out that from the 90s we will become01:17:18
could you quickly say so, that is, you are it01:17:20
everyone knows this is not a theorem you know this01:17:23
that 108 in half will be 90 that's what01:17:27
This is followed by another interesting fact01:17:29
it follows that if you take01:17:30
right triangle on and around01:17:32
describe him01:17:34
the circle will be the hypotenuse will be01:17:36
diameter i.e. hypotenuse and diameter01:17:38
this is the same diameter as described01:17:40
circles that's what's important and what's next01:17:43
one more property01:17:44
such a median is drawn from a right angle01:17:48
in a right triangle is equal to01:17:49
half the hypotenuse that is, if like this01:17:51
do it like this from a right angle01:17:53
from this angle draw the median she01:17:56
the diameter will be divided in half01:17:58
so that means she will come to the center01:18:01
circle to a point and therefore the most01:18:05
the median will be the radius, that is, this01:18:08
radius is radius is radius 3 radii01:18:12
this is how it follows from here that01:18:17
median is01:18:18
the hypotenuse is bisected because the median01:18:21
this is in fact the radius and the hypotenuse01:18:24
of this fact the diameter is actually here01:18:27
in this formula you know the radius is equal to01:18:29
diameter along the shafts, you know all this01:18:31
the radius is the diameter in half01:18:33
you know very well, so here in the theorem01:18:36
the same thing is written, just words01:18:38
replaced with other radius Diana a01:18:41
diameter to hypotenuse and01:18:43
and like this01:18:46
this is such a cool rem01:18:50
now the properties regarding the carried out01:18:52
from one point01:18:57
super-super-super understandable tower that is01:19:00
if you take it like this from one01:19:02
draw points for tangent here is one01:19:05
here's one of his other acacias won't01:19:08
equal actually it's not difficult01:19:10
believe me here I am through the center like this01:19:13
I’ll draw an axis of symmetry around the circle, yes01:19:15
here and here you can see that in its entirety01:19:18
symmetrical picture but difficult to do01:19:21
it's hard to believe that they are not equal01:19:24
we are talking about these segments, this one01:19:27
blue segment and this blue segment01:19:29
they are symmetrically uneven can prove01:19:32
that they are equal through the equality of these01:19:34
here are the first and second triangles01:19:36
rectangular firstly secondly here01:19:38
their radii are the same until thirdly01:19:40
they have a common country and theirs is rectangular01:19:42
triangle if there is at least strange and01:19:44
angle or even if they just have a leg and01:19:46
the hypotenuse is equal and they are all equal to one hundred01:19:49
percent is always because they01:19:52
there are already such things in one corner01:19:56
things are good they are equal01:19:59
some people still don't understand, wait01:20:02
they say let's go to this point01:20:04
We’ll just move it closer here then01:20:05
it will shorten relatively, look01:20:08
I'll explain here. here is a part here01:20:11
here regarding these here is one here01:20:13
another type, now they are equal, but this01:20:16
you put it that way, yes you can say01:20:18
let's take this point and go there01:20:19
let's move it closer here, closer then01:20:23
there will be 1 tangent in short I want01:20:26
say but it's not like that, watch this and01:20:29
it will be evidently shorter but different01:20:31
it will also become shorter, it will only come soon01:20:33
such01:20:35
because the point of contact will shift01:20:38
because there the point has moved, you see01:20:40
fierce now believe and she immediately eyes01:20:45
it is clear that they are equal now the following is01:20:48
inscribed and circumscribed quadrilateral01:20:52
inscribed it means inside01:20:56
inside01:20:57
described means he is around that is, after all, in01:21:01
inside about about01:21:06
described well yes against and logical01:21:09
that is, it means like this01:21:11
quadrangle outside01:21:14
Now01:21:16
question guys, look at the acasta clan01:21:19
any quadrilateral can be inscribed01:21:21
write only special not any but01:21:25
triangles are not the case, the thing is that01:21:27
near any triangle you can end01:21:28
write a circle therefore does not exist01:21:33
anyone what's the matter why so because01:21:36
to describe around a triangle01:21:38
circling you need to guide him through three01:21:40
points and through any three points you can01:21:42
you can describe a circle01:21:44
imagine such such mental01:21:45
experiment here take a soap bubble01:21:47
then put it on these two dots01:21:49
inflate inflate inflate put on01:21:51
and he will gradually inflate to such an extent01:21:53
image that is not ready. will capture understood01:21:57
that is, in any case 3. will take over01:21:59
this circle01:22:00
so through any three points you can01:22:03
draw a circle and only one01:22:06
the same applies to inscribed 401:22:09
inscribed in a triangle of a circle in01:22:12
any triangle turns out to be possible01:22:14
inscribe a circle in any absolutely01:22:16
maybe like this, soapy01:22:19
put a bubble there, come on, but01:22:21
let's but come on and he will sooner or later01:22:23
will run into the third side, what does it have to do with it01:22:25
three . yes you see three dots, that is, you can01:22:28
was to prove through01:22:30
theorem that through three points you can01:22:33
carry only one load and here01:22:36
would work well too01:22:38
in general, you can go into any triangle01:22:41
write describe the circle for this it01:22:43
no special features needed but but01:22:48
4 square only special only in01:22:53
special can be entered and described01:22:55
he must have such characteristics01:22:59
cyclic quadrilateral should01:23:02
the peculiarity is that he has01:23:03
these opposite angles are red01:23:06
and green ones, let’s say, add up to 180 or01:23:10
the other two opposite ones will guess the sum01:23:13
18001:23:14
where did it come from it's what you need01:23:18
consider a quadrilateral in no way01:23:20
quadrilateral and how the two corners of the frame go01:23:22
you lean on a friend in your green corner01:23:24
rests on the b.d. arc. on arc b d01:23:29
leans and red is blocked on the arc01:23:32
gd red another red and green arc01:23:36
the sum gives 360 degrees, but we and01:23:39
we know that inscribed angles are01:23:41
half an arc is their sum01:23:44
the sums of these arcs01:23:47
it's 360 in half because it's not accurate01:23:52
look01:23:53
[music]01:23:57
Now01:24:00
let's say this is an arc this is y this arc is01:24:05
x we have our inscribed angles are y01:24:08
in half and x in half so and because01:24:11
angles are halves of an arc01:24:15
Here01:24:18
that is, we mean that y plus x is 36001:24:22
to find out what their sum is equal to01:24:25
halves you just need this expression01:24:27
entirely in the west is it not here on01:24:29
in fact it turns out, let's do it like this01:24:33
we just use this fraction01:24:35
let's connect that is x plus y plus x in half01:24:39
you can also use a common denominator01:24:40
make a weight sexy 360 360 divide by01:24:43
2 is 180 that's where 180 comes from because01:24:48
the sum of 360 inscribed angle is half01:24:51
debt therefore 360 divided by 2 180 here01:24:54
where does this property come from now01:24:56
described quadrilateral quadrilateral01:24:59
can be described firstly only in01:25:02
a special quadrilateral can be inscribed01:25:04
circle01:25:05
I started talking they are special they have01:25:08
some kind of peculiarity and this is what it is01:25:10
the peculiarity turns out to be that the sum of the sides01:25:13
opposites must be equal01:25:16
why do you remember telling you that01:25:19
if you draw from one point01:25:23
tangents then they will be equal, that is01:25:25
then these two little ones cut off what01:25:27
they are equal he said to himself01:25:29
asymmetrical line asymmetrical01:25:31
picture there to believe and not to believe that's all01:25:33
I'll explain this here we can01:25:36
take these two segments as equal01:25:38
letters c small small give the same01:25:42
we will do the same with other parties01:25:44
other vertices, for example, we have an angle01:25:46
here too one o2 tangent01:25:49
greenies let it be b&b why01:25:52
because they are the same so you can01:25:53
take it by one letter the same with01:25:55
angle and they are the same segments a.a.01:25:59
and here is the same yes and now quickly01:26:03
write down a.b. what is this a + b a + b01:26:08
this is plus b01:26:11
plus b01:26:13
this is a + b to c yes this is what this is c +01:26:19
d em look at picture c + d01:26:22
bc is b + c respectively and a.d. this is a01:26:27
+ a01:26:29
+ d01:26:31
now look here four letters abcd01:26:35
and here the same thing a b c d just c01:26:39
in a different order01:26:41
it's all the same left and right and01:26:43
these quantities are equal, that's why it's like this01:26:48
property in such a focus you saw that is01:26:51
eat and describe these sides they consist of01:26:54
from identical pieces of here we are01:26:56
painted and got that it’s one and the same01:26:58
they're just mixed together, that's why01:27:01
such a property01:27:03
Fine01:27:05
now the angle between the tangent and the chord01:27:08
extremely rare formula but it does occur01:27:11
in the first part what are we talking about here if you01:27:14
put the chord here like this and if01:27:18
You want draw through the same point01:27:20
tangent if you want to find the angle between01:27:22
this proud and tangent is what you need01:27:26
let's say this arc is here01:27:28
100 degrees from this arc yes there it is necessary01:27:31
simply divide this arc by 2 and it will be 5001:27:33
that is, divide by 250 like this01:27:37
such a theorem can be done without this series01:27:40
you can actually get by because01:27:42
look01:27:44
I’ll quickly show you the hard drive and here it is01:27:47
chords you are given that this is 100 this is what01:27:51
The 100th can be seen here01:27:52
isosceles triangle this is 1 and y01:27:57
its angle is 100 because the central01:27:59
rests on an arc in this isosceles01:28:01
find these 2 triangles01:28:02
Hello because they are the same amount01:28:05
all angles of triangles 180 are shorter than 18001:28:08
minus 100 is 80 means every century01:28:10
this is 4040 I'm right here we have a right angle01:28:12
because this is the radius drawn in01:28:14
point of tangency here right angle here01:28:16
here 90 90 times 40 is 50 so the answer is01:28:20
50 that is, here it can be shorter in another way01:28:22
I can do without this formula, but here it is01:28:26
also very useful in some tasks01:28:29
now the property of chords is to be explained here in particular01:28:32
nothing, it’s just a property of chords that’s needed01:28:35
of her thousands he will learn to use it01:28:38
you just need the first part of this01:28:40
present the form01:28:42
numbers and everything and you get the answer01:28:45
desirable to remember because the first01:28:47
Sometimes I encounter this problem01:28:49
if you want to get an A there, get an A01:28:52
do you want to understand where this formula comes from?01:28:54
I assume you can consider these01:28:56
they have similar triangles like these01:28:58
the angles are equal would like to be equal too because01:29:00
that vertically you are inscribed here01:29:03
You can write down the proportion here and01:29:06
Even the most difficult thing will be possible to discharge you01:29:09
if through a similar triangle and01:29:11
write it down but for all the other guys I01:29:14
I'm just jealous, just remember everything01:29:16
in some tasks you can simply01:29:18
know the formula for which tax is not01:29:20
given01:29:22
here are a couple more forms that science01:29:25
not given01:29:27
but you need to know them if anything because01:29:30
they meet01:29:31
meets01:29:33
meets meets them task01:29:36
keep in mind01:29:38
Well, the area of the area of the figures is the area01:29:42
parallelogram01:29:43
for example, friends are actually one01:29:47
of the most important topics is geometry and many01:29:49
don’t understand how to find the area correctly01:29:51
but many people don’t understand what the frequency is01:29:54
this is what we really need to do here01:29:56
understand what base height is01:29:58
basis01:30:00
it is not difficult to understand, see the basis01:30:03
this is what is underneath01:30:05
many when ask schoolchildren01:30:08
says based on this is the country that01:30:10
is on the bottom side on the bottom01:30:13
this is your country01:30:15
but the funny thing is that either side can01:30:17
be on the bottom, that is, any side01:30:19
may be the basis01:30:21
for example, I flogged the gram and01:30:23
it follows from this that you01:30:25
the same area program actually01:30:27
In fact, you can write it in two ways01:30:30
here is the height of your base01:30:33
carried to this base, that is, this01:30:35
and right on the icon and this is on one side with01:30:38
on the other hand you can take the base01:30:43
b and01:30:44
here is the base b you can take under it01:30:47
there will be a different height no not the same01:30:50
because the height I again not everyone should01:30:52
be perpendicular, that is, h and01:30:54
we must be perpendicular to the base01:30:56
must be carried out precisely from01:30:58
the base, that is, the height will be01:31:01
something like this or something like this01:31:04
that is, it goes perpendicular to01:31:07
highest point01:31:09
Fine01:31:11
that is, you can write it another way01:31:14
the area you are looking at is possible01:31:16
imagine what exactly side b is01:31:18
is located at the bottom, that is, you turn it01:31:20
head ya country would be below from01:31:23
you go vertically up it01:31:24
do not stick perpendicular ones into the very01:31:26
the highest point of the program and this will be01:31:29
even with the icon b this height is also hers too01:31:33
can be used to calculate01:31:34
squares and look01:31:36
shape yes yes these are the bases to the height01:31:38
but it can be done in two different ways01:31:40
please keep it in your possession01:31:42
head forever01:31:45
Now01:31:47
here there is also a second formula via01:31:49
sinus01:31:50
it's actually the same formula01:31:55
as well as through the height, just here the height01:31:58
painted through the sine in a triangle01:32:00
a bh01:32:02
here, but let me show you the sine of the angle, what if01:32:05
paint it with a sign01:32:07
right up to this divided by b actually01:32:10
in fact, that is, the opposite leg01:32:12
divide by the hypotenuse if there is already01:32:14
express then it will be h equals01:32:17
seamus a multiplied by b and in fact here01:32:20
here it’s just this shape instead of height01:32:23
substituted it turned out another formula here01:32:26
keep in mind that this sine here is between01:32:28
two countries01:32:30
the city is about to country a country b a forces01:32:34
there must always be further between them01:32:37
area of triangle later triangle01:32:40
this is the same area of the parallelogram01:32:42
just divided by two because this one01:32:43
area of a parallelogram this shape01:32:46
square program this form of square01:32:47
the program was also simply divided into 2 more01:32:50
from above why so because programs01:32:52
if you take it like this and divide by 2 then01:32:54
it turns out to be a triangle, that's all01:32:57
when you are looking for the area of a triangle you01:33:00
first press the base to the height you01:33:01
find the area of this01:33:03
parallelogram and then divide by 201:33:05
because you only have this triangle01:33:07
you need to go only half of the program01:33:09
that's good now further square01:33:14
The rhombus area of the frame is also located as01:33:16
his programs because rum is the rule01:33:18
gram that is, we have if the program01:33:20
was01:33:21
there was a formula that there were two sides01:33:23
multiply by each other not much with between01:33:25
them is the sine of the angle, otherwise here because of that01:33:28
that the sides are equally just this this and01:33:32
this turns out to be squared here and01:33:34
it turns out this formula is this formula01:33:36
area program where it comes from here01:33:38
this formula is obtained from the formula d01:33:41
1 multiply by 2 divide by 2 multiply01:33:43
by the sine of the angle between the diagonals01:33:45
there is a formula that works for01:33:48
any quadrilateral you need it for01:33:50
you don’t need to know, but it’s just there01:33:53
for example in the game, that is, there is d1 d201:33:56
sine of the angle between them and just a joke in01:33:58
that01:34:00
here is this corner corner 1 yes here it is01:34:04
worth that is, it is the sine of the angle sine of the angle01:34:06
one funny thing is that here we have01:34:10
diagonally at an angle of 90 degrees01:34:12
motives at the beginning of the lesson in the middle of the lesson01:34:14
said that sine 90 is units even in01:34:16
the tables say so, so it turns out01:34:18
that is, one is taken as one and multiplied01:34:21
we hope it's pointless01:34:23
So all that remains is this form01:34:24
Rob has this formula here01:34:28
he just lost his sinus01:34:31
because they are really here01:34:34
what are the corners and what did I do oh my god01:34:36
everything is fine about01:34:39
ok I mean yeah it's between01:34:42
the diagonals are perpendicular because of this01:34:44
a lot of interesting things are happening01:34:47
this is just the area of the trapezoid it is given01:34:50
there is no need to memorize for abundance01:34:53
just know how to use01:34:55
now the properties of Diana and this property01:34:59
median is also a tax and is not given, but in01:35:03
In the second part, you often need to be able to01:35:06
paint the area and01:35:08
knowledge of this theorem really helps01:35:11
actually solve 2 parts and use01:35:13
you can’t need it exactly though why01:35:17
you can't of course you can of course you can01:35:19
on the same gives his textbooks01:35:20
can I watch it, you know01:35:23
that the median bisects the side of an angle01:35:26
it turns out to be half opposite01:35:28
divides the area in half, that is, the area01:35:30
1 triangle 2 are equal how come01:35:33
the fact is that the basis is01:35:37
are the same and the heights are the same01:35:40
triangles therefore do not have the same area01:35:43
will be the same by the way01:35:45
heights, many people need this01:35:47
was to begin, of course, to promise squares01:35:49
many people don't understand what height is01:35:51
triangle01:35:52
when you ask the height of a triangle01:35:55
Normally there is no problem lowering the height01:35:57
here is what height many people think01:36:00
that the height is carried out like this, but this01:36:03
your triangle is wrong then it is higher01:36:07
it goes further, so here you drop it01:36:09
such a small height is needed further01:36:12
pierce this ceiling and measure the height01:36:15
to the highest point this is the height01:36:17
such but of course mathematics is ordinary and01:36:20
Merritt not just like that but from the top01:36:23
released and lowered onto01:36:26
the continuation of the foundation will be like this01:36:29
correct and01:36:30
many schoolchildren do not understand what01:36:32
such when then the basis for what will be01:36:34
anyway guys, the basis is always the same01:36:36
the same base because up side a01:36:38
height01:36:39
the height is like this, that is, let’s say if01:36:43
this country is three and the height is up to 6 then01:36:47
this is where 3 is multiplied by 601:36:49
here you need to find the area, that is01:36:51
it will be 901:36:53
why so guys this is super logical01:36:56
look, imagine you live in01:36:58
this little one in one small room01:37:00
you need to measure the height of your ceiling01:37:03
you won’t measure all the way here01:37:05
so yours is already a higher room and that’s why01:37:08
you have to pierce the ceiling not to measure01:37:09
higher but you can’t get it that way so select01:37:12
from the street, that is, you go outside01:37:14
Yatsen high point to death01:37:16
this is the distance to the ground01:37:17
the height of your ceiling and01:37:19
in real life look like this01:37:22
The Leaning Tower of Pisa stands like this01:37:24
at an angle so you think and the height you need01:37:27
measure if the height is approximately01:37:31
that's how it will be a long tower they01:37:33
height height is how high on the01:37:36
ground and therefore you need to look at the level01:37:39
ground and from this ground level01:37:41
measure perpendicularly to the01:37:43
high point like this this is how it will be01:37:45
height they are01:37:46
height are perpendicular and up to01:37:49
high point01:37:51
Fine01:37:53
property of medians01:37:55
used in the first and second parts01:37:57
also very good and often01:38:01
pretty but not about him they forget about this01:38:04
she theorem01:38:06
very unusual actually look01:38:10
it turns out if the median is crossed between01:38:12
yourself01:38:13
the part that is near the top is twice01:38:16
longer than the rest of the part that is if01:38:19
here 10 let's say it turns out here01:38:22
will sing twice as long, that is01:38:24
ratio two to one two to one at01:38:28
solving the problem, this can be written down here01:38:30
so look01:38:31
you can say like let let a and this is 201:38:36
x then01:38:38
then a 1 is 1 x like this you can01:38:43
write down01:38:44
when solving problems this is such a lifehack01:38:47
which helps solve the problem easier01:38:49
then you put it there somehow01:38:52
there you can at least make an equation01:38:54
let’s say if you are given everything in its entirety01:38:56
the median is 15 until then you have 1501:39:00
equals 2x + x such a solver equation and01:39:04
you get the answer ok what else01:39:08
many people don’t know but it’s an interesting fact01:39:11
turns out to be the intersection point of the medians01:39:13
this center of gravity of the triangle is01:39:15
if you have a triangular plate01:39:16
swipe outside the median put there01:39:19
finger like this and you have this one01:39:21
the triangle will maintain balance01:39:22
because you will have it.01:39:25
center of gravity01:39:26
very interesting then01:39:29
property of a bisector, that is, we are on01:39:31
At this stage we went through all the geometry01:39:34
seriously without a day and this property01:39:39
only needed in room 25 in room 25 too01:39:43
everything we went through before and this01:39:45
property additionally this property01:39:47
bisectors01:39:48
turns out to be the bisector of the angle01:39:49
triangle divides the opposite01:39:51
side of the relationship length adjacent01:39:52
sides what this means can be learned here01:39:55
so look at the picture01:39:57
see l m divided by m n equals l n01:40:02
divide by n.p. if it's so inconvenient it's okay01:40:04
differently take another when here01:40:07
triangle01:40:08
so here you go01:40:10
mnp here we have a bisector01:40:16
these red sides of a friend's brother01:40:19
lose that is, lm divided by l.n. And01:40:22
these red faces are facing each other01:40:24
divide mb divide by n.p. So01:40:27
It’s easier to study with Victor like this too01:40:30
right and so right and so01:40:31
Right01:40:32
that's it, all the geometry is finished01:40:36
congratulations to you01:40:37
almost01:40:40
false start Thales' theorem still exists01:40:43
it is needed for task number 24 and number 2501:40:47
happens often and it’s very simple01:40:49
look if these segments are equal if01:40:53
parallel lines are these01:40:56
the segments are also equal it is used01:40:58
for example problem number 24 sometimes happens01:41:00
such a situation when you have a trapezoid01:41:03
in the middle line you get three01:41:04
parallel lines just like that01:41:06
here and01:41:07
it’s clear that the middle line divides01:41:09
this segment is in half side and this too01:41:11
divides it in half, it’s clear, but here’s how01:41:13
prove that the height is also halved01:41:15
Thales' theorem if you have segments here01:41:19
equal if parallel segments01:41:21
parallel lines cut off equal01:41:23
segments mean on another line too01:41:25
is equal to the segment from the secant, so these are01:41:28
the pieces of height are also equal01:41:30
here we need to lose useful for example but01:41:33
quite rare but that's all they are expensive01:41:36
friends, we went through all the geometry that01:41:39
need tax and of course it’s possible01:41:42
there was something you didn’t understand, because it was01:41:43
everything is very fast speedrun all that without01:41:46
examples especially without solving problems of course01:41:49
it’s so difficult, I understand you, I would at all01:41:51
I probably wouldn't understand anything if I looked01:41:53
this video seems to me01:41:54
so solve problems by type passed01:41:59
a piece of theory on solving problems passed01:42:02
a piece of theory on solving problems in this01:42:05
Of course, the file that exists will help you01:42:08
follow the link to this video01:42:10
use and enjoy the preparation01:42:13
everything will work out for you, I believe you, I hope01:42:16
you will be able to prepare and pass perfectly01:42:19
the exam was all your success with you01:42:21
mathematics teacher lg danir01:42:24
bye byeDescription:
🧩 ВСЕ ТИПЫ ЗАДАЧ, КОТОРЫЕ БУДУТ НА ОГЭ, В ОДНОМ ФАЙЛЕ Забирай ВК 👉🏻 https://vk.cc/clzuh0 В телеграм-канале в закрепе 👉🏻 https://t.me/+SyWH_LJ8GbCw5Dfn 📒 Вся геометрия в полном файле 👉🏻 https://vk.cc/clzuVr 🎁 Заходи на сайт и получи пробный курс в подарок: https://umschool.net/ Хочешь записаться на курс? Переходи по ссылке, чтобы получить цены и узнать подробнее — https://t.me/mathdanirBot?start=video Проводит занятие - Данир Баев, преподаватель по математике ОГЭ в онлайн-школе Умскул. - Данир преподает 5 лет - Выпускник МФТИ - Выпустил более 3000 учеников - Каждый третий ученик сдал ОГЭ на 5 в 2021 году Отзывы о мастер-группе: https://vk.com/topic-168456727_42034586 Телеграм-канал - https://t.me/danirmath Группа ВКонтакте - https://vk.me/umschmathoge Подписывайся на рассылку, чтобы каждую неделю получать полезные материалы для подготовки к ОГЭ по математике и приглашения на вебинары - https://ndlr.cc/huGgl6gz4 Нужна помощь или консультация? Позвоните нам, мы поможем! 8 800 300 63 24 (звонок бесплатный из любой точки РФ) Подписывайся на канал, ставь лайк и колокольчик, чтобы не пропустить новые вебинары 0:00:00 — Для чего этот ролик 0:00:17 — Как понять геометрию 0:00:42 — Все задачи ОГЭ в одном файле 0:01:10 — Разбор всех задач ОГЭ за 7 дней — курс Предбанник 0:04:02 — Всё про углы 0:07:00 — Вертикальные углы 0:08:37 — Виды треугольников 0:07:45 — Равнобедренный треугольник 0:13:14 — Равносторонний треугольник 0:14:03 — Элементы равностороннего треугольника 0:15:53 — Медиана, биссектриса и высота 0:16:25 — Признаки равенства треугольников 0:21:28 — Признаки параллельности 0:28:46 — Четырёхугольники 0:31:14 — Свойства параллелограмма 0:34:14 — Свойства прямоугольника и квадрата + ромба 0:36:09 — Свойства трапеции + средняя линия трапеции 0:39:00 — Прямоугольный треугольник. Теорема Пифагора 0:41:36 — Sin, cos, tg, ctg острых углов 0:47:45 — Табличные значения Sin, cos, tg, ctg 0:49:17 — Выпендрежная таблица значений Sin, cos, tg, ctg 0:50:39 — sin 90 0:52:17 — Sin, cos, tg, ctg тупых углов 0:52:58 — Тригонометрические тождества 0:57:34 — Теорема синусов 0:59:19 — Теорема косинусов 1:02:08 — Подобные треугольники. Признаки подобия 1:07:33 — 4 случая подобия в ОГЭ 1:11:41 — Площадь и периметр подобных треугольников 1:13:54 — Вписанный и центральный угол 1:14:53 — Свойства вписанного и центрального угла 1:16:38 — Теорема об угле, опирающемся на диаметр окружности 1:18:50 — Свойства касательных, проведённых из одной точки 1:20:48 — Вписанные и описанные четырёхугольники 1:22:49 — Cвойства вписанного и описанного четырёхугольника 1:26:06 — Угол между касательной и хордой 1:28:30 — Свойства: хорд, секущих, секущей и касательной 1:29:39 — Площади фигур 1:34:56 — Свойство медианы 1:37:54 — Свойство медиан 1:39:29 — Свойство биссектрисы 1:43:55 — Свойство медианы 1:40:40 — Теорема Фалеса 1:41:35 — Завершение
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