Lesson topic
Lesson Objectives
- Get acquainted with new definitions and remember some already studied.
- Deepen your knowledge of geometry, study the history of origin.
- To consolidate students' theoretical knowledge about triangles in practical activities.
- Introduce students to the Egyptian triangle and its use in construction.
- Learn to apply the properties of shapes when solving problems.
- Developmental – to develop students’ attention, perseverance, perseverance, logical thinking, mathematical speech.
- Educational - through the lesson, cultivate an attentive attitude towards each other, instill the ability to listen to comrades, mutual assistance, and independence.
Lesson Objectives
- Test students' problem-solving skills.
Lesson Plan
- Opening remarks.
- It's useful to remember.
- Toegon.
Opening remarks
Did they know mathematics and geometry in ancient Egypt? They not only knew it, but also constantly used it when creating architectural masterpieces and even... during the annual marking of fields in which flood water destroyed all the boundaries. There was even a special service of surveyors who quickly, using geometric techniques, restored the boundaries of fields when the water subsided.
It is not yet known what we will call our younger generation, which grows up on computers that allow us not to memorize the multiplication table or perform other elementary mathematical calculations or geometric constructions in our heads. Maybe human robots or cyborgs. The Greeks called those who could not prove a simple theorem without outside help ignoramuses. Therefore, it is not surprising that the theorem itself, which was widely used in applied sciences, including for marking fields or building pyramids, was called by the ancient Greeks “the bridge of donkeys.” And they knew Egyptian mathematics very well.
Useful to remember
Triangle
Triangle rectilinear, a part of the plane limited by three straight segments (sides of the Triangle (in geometry)), each having one common end in pairs (vertices of the Triangle (in geometry)). A triangle whose lengths of all sides are equal is called equilateral, or correct, Triangle with two equal sides - isosceles. The triangle is called acute-angled, if all its angles are sharp; rectangular- if one of its angles is right; obtuse-angled- if one of its angles is obtuse. A triangle (in geometry) cannot have more than one right or obtuse angle, since the sum of all three angles is equal to two right angles (180° or, in radians, p). The area of the Triangle (in geometry) is equal to ah/2, where a is any of the sides of the Triangle, taken as its base, and h is the corresponding height. The sides of the Triangle are subject to the following condition: the length of each of them is less than the sum and greater than the difference in the lengths of the other two sides.
Triangle- the simplest polygon having 3 vertices (angles) and 3 sides; part of the plane bounded by three points and three segments connecting these points in pairs.
- Three points in space that do not lie on the same straight line correspond to one and only one plane.
- Any polygon can be divided into triangles - this process is called triangulation.
- There is a section of mathematics entirely devoted to the study of the laws of triangles - Trigonometry.
Types of Triangles
By type of angles
Since the sum of the angles of a triangle is 180°, at least two angles in the triangle must be acute (less than 90°). The following types of triangles are distinguished:
- If all the angles of a triangle are acute, then the triangle is called acute;
- If one of the angles of a triangle is obtuse (more than 90°), then the triangle is called obtuse;
- If one of the angles of a triangle is right (equal to 90°), then the triangle is called right-angled. The two sides that form a right angle are called legs, and the side opposite the right angle is called the hypotenuse.
According to the number of equal sides
- A scalene triangle is one in which the lengths of the three sides are pairwise different.
- An isosceles triangle is one in which two sides are equal. These sides are called lateral, the third side is called the base. In an isosceles triangle, the base angles are equal. The altitude, median and bisector of an isosceles triangle lowered to the base are the same.
- An equilateral triangle is one in which all three sides are equal. In an equilateral triangle, all angles are equal to 60°, and the centers of the inscribed and circumscribed circles coincide.
– a right triangle with an aspect ratio of 3:4:5. The sum of these numbers (3+4+5=12) has been used since ancient times as a unit of multiplicity when constructing right angles using a rope marked with knots at 3/12 and 7/12 of its length. The Egyptian triangle was used in the architecture of the Middle Ages to construct proportional schemes.
So where to start? Is it because of this: 3 + 5 = 8. and the number 4 is half the number 8. Stop! The numbers 3, 5, 8... Don't they resemble something very familiar? Well, of course, they are directly related to the golden ratio and are included in the so-called “golden series”: 1, 1, 2, 3, 5, 8, 13, 21
... In this series, each subsequent term is equal to the sum of the previous two: 1 + 1= 2. 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8
and so on. It turns out that the Egyptian triangle is related to the golden ratio? And did the ancient Egyptians know what they were dealing with? But let's not rush to conclusions. It is necessary to find out more details.
The expression “golden ratio”, according to some, was first introduced in the 15th century Leonardo da Vinci
. But the “golden series” itself became known in 1202, when the Italian mathematician first published it in his “Book of Counting” Leonardo of Pisa
. Nicknamed Fibonacci. However, almost two thousand years before them, the golden ratio was known Pythagoras and his students. True, it was called differently, as “division in the average and extreme ratio.” But the Egyptian triangle with its The “golden ratio” was known back in those distant times when the pyramids were built in Egypt when Atlantis flourished.
To prove the Egyptian triangle theorem, it is necessary to use a line segment of known length A-A1 (Fig.). It will serve as a scale, a unit of measurement, and will allow you to determine the length of all sides of the triangle. Three segments A-A1 are equal in length to the smallest side of triangle BC, whose ratio is 3. And four segments A-A1 are equal in length to the second side, whose ratio is expressed by the number 4. And, finally, the length of the third side is equal to five segments A -A1. And then, as they say, it’s a matter of technique. On paper we will draw a segment BC, which is the smallest side of the triangle. Then, from point B with a radius equal to the segment with ratio 5, we draw a circular arc with a compass, and from point C, an arc of a circle with a radius equal to the length of the segment with ratio 4. If we now connect the intersection point of the arcs with lines to points B and C, we obtain a right triangle aspect ratio 3:4:5.
Q.E.D.
The Egyptian triangle was used in the architecture of the Middle Ages to construct proportionality schemes and to construct right angles by surveyors and architects. The Egyptian triangle is the simplest (and first known) of the Heronian triangles - triangles with integer sides and areas.
Egyptian triangle - a mystery of antiquity
Each of you knows that Pythagoras was a great mathematician who made invaluable contributions to the development of algebra and geometry, but he gained even more fame thanks to his theorem.
And Pythagoras discovered the Egyptian triangle theorem at the time when he happened to visit Egypt. While in this country, the scientist was fascinated by the splendor and beauty of the pyramids. Perhaps this was precisely the impetus that exposed him to the idea that some specific pattern was clearly visible in the shapes of the pyramids.
History of discovery
The Egyptian triangle received its name thanks to the Hellenes and Pythagoras, who were frequent guests in Egypt. And this happened approximately in the 7th-5th centuries BC. e.
The famous pyramid of Cheops is actually a rectangular polygon, but the pyramid of Khafre is considered to be the sacred Egyptian triangle.
The inhabitants of Egypt compared the nature of the Egyptian triangle, as Plutarch wrote, with the family hearth. In their interpretations one could hear that in this geometric figure its vertical leg symbolized a man, the base of the figure related to the feminine principle, and the hypotenuse of the pyramid was assigned the role of a child.
And already from the topic you have studied, you are well aware that the aspect ratio of this figure is 3: 4: 5 and, therefore, that this leads us to the Pythagorean theorem, since 32 + 42 = 52.
And if we take into account that the Egyptian triangle lies at the base of the Khafre pyramid, then we can conclude that the people of the ancient world knew the famous theorem long before it was formulated by Pythagoras.
The main feature of the Egyptian triangle was most likely its peculiar aspect ratio, which was the first and simplest of the Heronian triangles, since both the sides and its area were integers.
Features of the Egyptian Triangle
Now let's take a closer look at the distinctive features of the Egyptian triangle:
First, as we have already said, all its sides and area consist of integers;
Secondly, by the Pythagorean theorem we know that the sum of the squares of the legs is equal to the square of the hypotenuse;
Thirdly, with the help of such a triangle you can measure right angles in space, which is very convenient and necessary when constructing structures. And the convenience is that we know that this triangle is right-angled.
Fourthly, as we also already know, even if there are no appropriate measuring instruments, this triangle can be easily constructed using a simple rope.
Application of the Egyptian triangle
In ancient centuries, the Egyptian triangle was very popular in architecture and construction. It was especially necessary if a rope or cord was used to build a right angle.
After all, it is known that laying a right angle in space is quite a difficult task, and therefore enterprising Egyptians invented an interesting way of constructing a right angle. For these purposes, they took a rope, on which they marked twelve even parts with knots and then folded a triangle from this rope, with sides that were equal to 3, 4 and 5 parts and in the end, without any problems, they got a right triangle. Thanks to such an intricate tool, the Egyptians measured the land with great precision for agricultural work, built houses and pyramids.
This is how a visit to Egypt and studying the features of the Egyptian pyramid prompted Pythagoras to discover his theorem, which, by the way, was included in the Guinness Book of Records as the theorem that has the largest amount of evidence.
Triangular Reuleaux wheels
Wheel- a round (as a rule), freely rotating or fixed on an axis disk, allowing a body placed on it to roll rather than slide. The wheel is widely used in various mechanisms and tools. Widely used for transporting goods.
The wheel significantly reduces the energy required to move a load on a relatively flat surface. When using a wheel, work is performed against the rolling friction force, which in artificial road conditions is significantly less than the sliding friction force. Wheels can be solid (for example, a wheel pair of a railway carriage) and consisting of a fairly large number of parts, for example, a car wheel includes a disk, rim, tire, sometimes a tube, fastening bolts, etc. Car tire wear is almost a solved problem (if the wheel angles are set correctly). Modern tires travel over 100,000 km. An unsolved problem is the wear of tires on airplane wheels. When a stationary wheel comes into contact with the concrete surface of the runway at a speed of several hundred kilometers per hour, the tire wear is enormous.
- In July 2001, an innovative patent was received for the wheel with the following wording: “a round device used for transporting goods.” This patent was issued to John Kao, a lawyer from Melbourne, who thereby wanted to show the imperfections of Australian patent law.
- In 2009, the French company Michelin developed a mass-produced car wheel, the Active Wheel, with built-in electric motors that drive the wheel, spring, shock absorber and brake. Thus, these wheels make the following vehicle systems unnecessary: engine, clutch, gearbox, differential, drive and drive shafts.
- In 1959, the American A. Sfredd received a patent for a square wheel. It easily walked through snow, sand, mud, and overcame holes. Contrary to fears, the car on such wheels did not “limp” and reached speeds of up to 60 km/h.
Franz Relo(Franz Reuleaux, September 30, 1829 - August 20, 1905) - German mechanical engineer, lecturer at the Berlin Royal Academy of Technology, who later became its president. The first, in 1875, to develop and outline the basic principles of the structure and kinematics of mechanisms; He dealt with the problems of aesthetics of technical objects, industrial design, and in his designs attached great importance to the external forms of machines. Reuleaux is often called the father of kinematics.
Questions
- What is a triangle?
- Types of triangles?
- What is special about the Egyptian triangle?
- Where is the Egyptian triangle used? > Mathematics 8th grade
Construction using the Egyptian triangle is an ancient method that is still actively used by modern builders. It got its name thanks to ancient Egyptian buildings, although it is known that its history begins long before this period.
But, most likely, the properties of the unique figure were not appreciated in those days until Pythagoras appeared, who was able to analyze and evaluate the graceful forms of the figure.
The Egyptian triangle has been known since ancient times. It has been and remains popular in construction and architecture for many centuries.
It is believed that the great Greek mathematician Pythagoras of Samos created the geometric structure. Thanks to him, today we can use all the properties of geometric construction in the field of structure.
The birth of an idea
The mathematician got the idea after traveling to Africa at the request of Thales, who set the task for Pythagoras to study the mathematics and astronomy of those places. In Egypt, among the endless desert, he encountered majestic buildings that amazed him with their size, grace and beauty.
It should be noted that more than two and a half thousand years ago the pyramids were somewhat different - huge, with clear edges. Having carefully studied the powerful buildings, of which there were quite a few, since next to the giants there were smaller temples built for the children, wives and other relatives of the pharaoh, this gave him an idea.
Thanks to his mathematical abilities, Pythagoras was able to determine the pattern in the shapes of the pyramid, and his ability to analyze and draw conclusions led to the creation of one of the most significant theories in the history of geometry.
From history
Did they know about geometry and mathematics in ancient Egypt? Of course yes. The life of the Egyptians was closely connected with science. They regularly used their knowledge when marking fields and creating architectural masterpieces. There was even a service of land surveyors who applied geometric rules when restoring boundaries.
The triangle received its name thanks to the Hellenes, who often visited Egypt in the 7th-5th centuries. BC It is believed that the prototype of the figure was Cheops pyramid, characterized by perfect proportions. Her place in history is special. If you look at the cross section, you can see two triangles, whose internal angle is 51 about 50’.
Structure
The task is much easier if you use a protractor or triangle. But, previously only cords and ropes, divided into segments, were used. Thanks to the marks on the rope, it was possible to accurately recreate a rectangular figure. The builders replaced the protractor and square with a rope, for which they marked 12 parts with knots on it and folded a triangle with segments 3,4,5. A right angle was obtained without difficulty. This knowledge helped create many structures, including the pyramids.
It is interesting that before ancient Egypt, they built in this way in China, Babylon, and Mesopotamia.
The properties of the Egyptian triangular figure obey the truth - the square of the hypotenuse is equal to the squares of the two legs. This Pythagorean theorem is familiar to everyone from school. For example, we multiply 5x5 and get a hypotenuse equal to the number 25. The squares of both sides are 16 and 9, which adds up to 25.
Thanks to these properties, the triangle has found application in construction. You can take any part in order to draw a straight line with the condition that its length must be a multiple of five. After this, notice one edge and draw a line from it that is a multiple of four, and from the other a line that is a multiple of three. In this case, each segment must be at least four and three in length. Intersecting, they form one right angle of 90 degrees. Other angles are 53.13 and 36.87 degrees.
What alternatives are there?
How to create a right angle
The best option make a right angle is the use of a square or protractor. This will allow you to find the required proportions with minimal cost. But, the main point of the Egyptian triangle is its versatility due to the ability to create a figure without having anything at hand.
Anything can be useful in this matter, even printed publications. Any book or even magazine always has an aspect ratio that forms a right angle. Printing presses always work precisely so that the roll inserted into the machine is cut at proportional angles.
Ancient engineers came up with many ways to build the Egyptian triangle and always saved resources.
Therefore, the simplest and most widely used method was the method of constructing a geometric figure using ordinary rope. The string was taken and cut into 12 even pieces, from which a figure with proportions of 3,4 and 5 was laid out.
How to create other angles?
The Egyptian Triangle cannot be underestimated in the construction world. Its properties are definitely useful, but without the ability to construct angles of a different degree in construction it is impossible. To form an angle of 45 degrees, you will need a frame or baguette, which is sawn at an angle of 45 degrees and connected to each other.
Important! To get the required slope, you will need to borrow a sheet of paper from the printed publication and bend it. The bend lines will go through the corner. The edges must be connected.
You can get 60 degrees using two 30 degree triangles. Most often used to create decorative elements.
Small tricks
The Egyptian triangle 3x4x5 is relevant for small houses. But what if the house is 12x15?
To do this, you need to construct a right triangle whose legs are 12 and 15 m. The hypotenuse is found as the square root of the sum of 12x12 and 15x15. As a result, we get 19.2 m. Using something - rope, twine, twine, cable, military cable, we measure 12, 15 and 19.2 m. We make knots in these places and put presses.
Then you need to stretch the triangle in the right place and install 3 support points into which to drive pegs. The fourth point can be obtained without touching the ends of the legs. To do this, the right angle point is thrown diagonally and everything is ready.
For example, there is an area where a right angle is required - for space for a kitchen unit, tile layout and other aspects. It would be nice to take such issues into account when laying, but the reality is different and you don’t always come across smooth walls and right angles. The Egyptian triangle with a ratio of 3:4:5, or, if necessary, 1.5:2:2.5, is useful here.
The thickness of the beacons, errors, bumps on the walls, etc. must be taken into account. The triangle is drawn using a tape measure and chalk. If the markings are small, then you can use a sheet, since they are cut with the correct angles.
The Egyptian triangle was widely used in construction for as long as 2.5 centuries. And today, sometimes it is necessary to use this technique, in the absence of the necessary tools, to obtain right angles. The properties of this figure are unique, which guarantees precision in architecture and construction, which cannot be avoided. It is easy to work with, its shape is harmonious and beautiful. To this day, inquisitive minds are trying to unravel the mystery of the Egyptian triangle.
Anyone who listened carefully to a geometry teacher at school is very familiar with what the Egyptian triangle is. It differs from other types of similar ones with an angle of 90 degrees in its special aspect ratio. When a person first hears the phrase “Egyptian triangle,” pictures of majestic pyramids and pharaohs come to mind. But what does history say?
As is always the case, there are several theories regarding the name "Egyptian Triangle". According to one of them, the famous Pythagorean theorem came to light precisely thanks to this figure. In 535 BC. Pythagoras, following the recommendation of Thales, went to Egypt in order to fill some gaps in his knowledge of mathematics and astronomy. There he drew attention to the peculiarities of the work of Egyptian land surveyors. They performed a construction with a right angle in a very unusual way, the sides of which were interconnected with one another in a 3-4-5 ratio. This mathematical series made it relatively easy to connect the squares of all three sides with one rule. This is how the famous theorem arose. And the Egyptian triangle is precisely the same figure that prompted Pythagoras to a most ingenious solution. According to other historical data, the figure was given its name by the Greeks: at that time they often visited Egypt, where they could be interested in the work of land surveyors. There is a possibility that, as often happens with scientific discoveries, both stories happened at the same time, so it is impossible to say with certainty who came up with the name “Egyptian triangle” first. Its properties are amazing and, of course, are not limited to the aspect ratio alone. Its area and sides are represented by integers. Thanks to this, applying the Pythagorean theorem to it allows us to obtain integer numbers of the squares of the hypotenuse and legs: 9-16-25. Of course, this could be just a coincidence. But how, in this case, can we explain the fact that the Egyptians considered “their” triangle sacred? They believed in his interconnection with the entire Universe.
After information about this unusual geometric figure became publicly available, the world began searching for other similar triangles with integer sides. It was obvious that they existed. But the importance of the question was not simply to perform mathematical calculations, but to test the “sacred” properties. The Egyptians, for all their unusualness, were never considered stupid - scientists still cannot explain how exactly the pyramids were built. And here, suddenly, an ordinary figure was attributed a connection with Nature and the Universe. And, indeed, the found cuneiform contains instructions about a similar triangle with a side whose size is described by a 15-digit number. Currently, the Egyptian triangle, whose angles are 90 (right), 53 and 37 degrees, is found in completely unexpected places. For example, when studying the behavior of molecules of ordinary water, it turned out that the change is accompanied by a restructuring of the spatial configuration of the molecules, in which you can see... that same Egyptian triangle. If we remember that it consists of three atoms, then we can talk about conditional three sides. Of course, we are not talking about a complete coincidence of the famous ratio, but the resulting numbers are very, very close to the required ones. Is this why the Egyptians recognized their “3-4-5” triangle as a symbolic key to natural phenomena and the secrets of the Universe? After all, water, as you know, is the basis of life. Without a doubt, it is too early to draw an end to the study of the famous Egyptian figure. Science never rushes to conclusions, seeking to prove its assumptions. And we can only wait and be amazed at the knowledge
Mathematical lifehack from the field of geometry “How to get a triangle with a right angle using a simple rope.”
The Egyptians 4,000 years ago used a method to build the pyramids by making a right triangle using a rope divided into 12 equal parts.
The concept of the “Egyptian triangle”.
Why is a triangle with sides 3, 4, 5 called Egyptian?
And the whole point is that the builders of Ancient Egypt pyramids needed a simple and reliable method for constructing a triangle with a right angle. And this is how they implemented it. The rope was divided into twenty equal parts, marking the boundaries between adjacent parts; the ends of the rope were connected. After this, 3 people pulled the rope so that it formed a triangle, and the distances between each two Egyptians pulling the rope were three parts, four parts and five parts respectively. The result was a triangle with a right angle with legs in three and four parts and a hypotenuse in five parts. It is known that the angle between sides of three and four parts was right. As you know, ancient Egyptian surveyors, who in addition to measuring land plots were engaged in construction on the ground, in ancient Egypt they were called harpedonaptes (which literally translates as “pulling ropes”). Harpedonaptes occupied 3rd place in the hierarchy of priests of Ancient Egypt.
Converse Pythagorean theorem.
But what makes a triangle with sides 3, 4, 5 turn out to be rectangular? Most would answer this question by saying that this fact is a theorem: since three squared plus four squared equals five squared. But he says that if a triangle has a right angle, then the sum of the squares of its 2 sides is equal to the square of the third. Here we are dealing with a theorem inverse to the Pythagorean theorem: if the sum of the squares of 2 sides of a triangle is equal to the square of the third, then the triangle is right-angled.
The practical application outlined goes back to the distant past. Hardly anyone gets right angles using this method today. But nevertheless, this method is an excellent mathematical life hack and can be applied by you in any life situation.
The method of determining a right triangle using a rope has moved from the world of practice to the world of ideas, just as much of the material culture of antiquity has entered the spiritual culture of present reality.
The Egyptian triangle and its properties have been well known since ancient times. This figure was widely used in construction for marking and constructing correct angles.
History of the Egyptian Triangle
The creator of this geometric design is one of the greatest mathematicians of antiquity, Pythagoras. It is thanks to his mathematical research that we can fully use all the properties of this geometric structure in construction.
It can be assumed that mathematical skills allowed Pythagoras to notice a pattern in the forms of the structure. The further development of events can be easily imagined. Basic analysis and drawing conclusions created one of the most significant figures in history. Most likely, it was the Cheops pyramid that was chosen as the prototype because of its almost perfect proportions.
Egyptian triangle in construction
The properties of this unique geometric structure are that its construction without the use of any tools allows you to build a house with angles that are correct in all relationships.
Important! Of course, ideally the best option would be to use a protractor or square.
So, the qualities of the Egyptian triangle allow you to make angles that are correct in all relationships. The sides of the structure have the following ratio to each other:
To check whether you have drawn the right figure, use the Pythagorean Theorem, well known from school.
Attention ! The properties of the Egyptian triangle are such that the square of the hypotenuse is equal to the squares of the two legs.
For a better understanding, let's take the above relationship and create a small example. Let's multiply five by five. As a result, we get a hypotenuse equal to 25. Let's calculate the squares of two legs. They will be 16 and 9. Accordingly, their sum will be twenty-five.
This is why the properties of the Egyptian triangle are so often used in construction. All you have to do is take the workpiece and draw a straight line. Its length should always be a multiple of 5. Then you need to mark one edge and measure a line divisible by 4 from it, and 3 from the second.
Attention ! The length of each segment will be 4 and 3 cm (at minimum values). The intersection of these lines forms a right angle equal to 90 degrees.
Alternative ways to construct a 90 degree right angle
As mentioned above, the best option is to simply take a square or protractor. These tools allow you to achieve the desired proportions with the least amount of time and effort. The main property of the Egyptian triangle is its versatility. A figure can be built with virtually nothing in your arsenal.
Simple printed materials help greatly in constructing a right angle. Take any magazine or book. The fact is that their aspect ratio is always exactly 90 degrees. Printing presses work very accurately. Otherwise, the roll that is fed into the machine will be cut at disproportionate crooked angles.
How to make an Egyptian triangle using a rope
The properties of this geometric figure are difficult to overestimate. It is not surprising that ancient engineers came up with many ways to form it using minimal resources.
One of the simplest is the method of forming the Egyptian triangle with all its attendant properties using a simple rope. Take the twine and cut it into 12 absolutely even pieces. From them, make a figure with proportions 3, 4 and 5.
How to construct an angle of 45, 30 and 60 degrees
Of course, the Egyptian triangle and its properties are very useful when building a house. But you still won’t be able to do without other angles. To get an angle of 45 degrees, take a frame or baguette material. Then cut it at an angle of forty-five degrees and join the halves to each other.
Important ! To obtain the desired slope, tear a piece of paper from the magazine and bend it. In this case, the bend lines will pass through the corner. The edges should match.
As you can see, the properties of the figure make it much easier and faster to build a geometric construct. To achieve an aspect ratio of 60 degrees, you need to take one triangle at 30º and the second the same. Typically, such proportions are necessary when creating certain decorative elements.
Attention ! A 30º aspect ratio is needed to make hexagons. Their properties are in demand in carpentry blanks.
Results
The properties of the Egyptian triangle have been widely used in construction for almost two and a half centuries. Even now, with a lack of tools, builders use this technique, discovered by Pythagoras, to achieve even right angles.
