Work on know-how in mathematics "incomparable likeness". Design and research work similarity of triangles in real life Research work similarity

the name of the project

Brief summary of the project

The project was prepared using design technology. It is implemented as part of the 8th grade geometry program on the topic "Triangle Similarity Signs". The project includes information and research part. Analytical work with information systematizes knowledge about similar figures. Independent research of students, as well as acquired practical knowledge, skills and abilities teach to see the importance of this theoretical material when applying it in practice. Didactic tasks will help to control the degree of assimilation of educational material.

Guiding questions

The fundamental question: "Does nature speak the language of likeness?"

“Is it possible to find examples of similarity around us?”, “How can I measure the height of my house?”, “Why do we need such triangles?”

Project plan

1. Brainstorming (formation of student research topics).

2. Formation of groups for research, hypotheses, discussion of ways to solve problems.

3.Choose a creative name for the project.

4. Discussion of the plan of theoretical and practical work of students in the group.

5. Discussion with students of possible sources of information.

6. Independent work of groups.

7. Preparation by students of presentations and reports on the report on the work done.

8. Presentation of research papers.

XXVanniversary city competition of educational and research
student work

Department of Education of the Administration of Kungur

Scientific Society of Students

section

Geometry

Kustova Ekaterina MAOU secondary school No. 13

8 "a" class

Supervisor:

Gladkikh Tatyana Grigorievna

MAOU secondary school №13

mathematic teacher

the highest category

Kungur, 2017

TABLE OF CONTENTS

Introduction………………………………………………………………………………3

Chapter 1

1.1. From the history of similarity …………………………………………………………….5

1.2. The concept of similarity ……………………………………………………………..6

1.3. Methods for measuring objects using similarity

1.3.1. The first way to measure the height of an object………………………….8

1.3.2. The second way to measure the height of an object………………………….9

1.3.3. The third way to measure the height of an object…………………………..11

2.1. Measuring the height of an object……………………………………………………..12

2.1.1. By the length of the shadow………………………………….. ………………………12

2.1. 2. With the help of a pole………………………………………………………………………13

2.1.3. With the help of a mirror……………………………………………………...13

2.1.4. How did the sergeant…………………………………………………...14

2.1.5. Without approaching the tree ……………………………………………….16

2.2. Pond cleaning. …………………………………………………...................................17

2.2.1. Methods for cleaning water bodies………………………………………………..17

2.2.2. Pond Width Measurement…………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………

Conclusion …………………………………………………………………… … ..22

References ………………………………………………………………...23



The semblance of beauty

Sometimes we don't notice

We say "Like a God"

Implying the ideal.



INTRODUCTION

The world in which we live is filled with the geometry of houses and streets, mountains and fields, the creations of nature and man. Geometry originated in ancient times. Building dwellings and temples, decorating them with ornaments, marking the ground, measuring distances and areas, a person applied his knowledge of the shape, size and relative position items derived from observation and experience. Almost all the great scientists of antiquity and the Middle Ages were eminent geometers. The motto of the ancient school was: "Those who do not know geometry are not allowed!"

Nowadays geometric knowledge still find wide application in construction, architecture, art, as well as in many industries. In geometry lessons, we studied the topic “Similarity of Triangles”, and I was interested in the question of how this topic can be applied in practice.

Remember the work of L. Carroll "Alice in Wonderland". What changes took place with the main character: either she grew to several feet, then decreased to several inches, always remaining, however, herself. What kind of transformation in terms of geometry are we talking about? Of course, about the transformation of similarity.

Objective:

Finding the area of ​​application of the similarity of triangles in human life.

Tasks:

1. Study the scientific literature on this topic.

2. Show the application of the similarity of triangles on the example of measuring work.

Hypothesis. Triangles can be used to measure real objects.

Research methods: search, analysis, mathematical modeling.

Chapter 1

1.1.From the history of similarity

The similarity of figures is based on the principle of ratio and proportion. The idea of ​​ratio and proportion originated in ancient times. This is evidenced by ancient Egyptian temples, details of the tomb of Menes and the famous pyramids at Giza (3rd millennium BC), Babylonian ziggurats (stepped cult towers), Persian palaces and other ancient monuments. Many circumstances, including the features of architecture, the requirements of convenience, aesthetics, technology and economy in the construction of buildings and structures, caused the emergence and development of the concepts of ratio and proportionality of segments, areas and other quantities. In the "Moscow" papyrus, when considering the ratio of a larger leg to a smaller one in one of the tasks for a right triangle, a special sign is used for the concept of "relationship". In Euclid's Elements, the doctrine of relationships is presented twice. Book VII contains the theory of arithmetic. It applies only to commensurate quantities and to whole numbers. This theory was created on the basis of the practice of working with fractions. Euclid uses it to study the properties of integers. Book V sets out general theory relations and proportions, developed by Eudoxus. It underlies the doctrine of the similarity of figures, set forth in the VI book of the "Beginnings", where the definition is found: "Similar rectilinear figures are those that have respectively equal angles and proportional sides."

Identical in shape, but different in size, figures are found in Babylonian and Egyptian monuments. In the surviving burial chamber of the father of Pharaoh Ramesses II, there is a wall covered with a network of squares, with the help of which smaller drawings were transferred to the wall in an enlarged form.

The proportionality of the segments formed on lines intersected by several parallel lines was known even to the Babylonian scientists. Although some attribute this discovery to Thales of Miletus. The ancient Greek sage Thales, six centuries before our era, determined the height of the pyramid in Egypt. He took advantage of her shadow. The priests and the pharaoh, gathered at the foot of the pyramid, looked puzzled at the northern stranger, who guessed the height of the huge structure from the shadow. Thales, says the legend, chose the day and hour when the length of his own shadow was equal to his height; at this point, the height of the pyramid must also equal the length of the shadow it casts.

A cuneiform tablet has survived to this day, in which we are talking on the construction of proportional segments by drawing parallels in a right triangle to one of the legs.

1.2. The concept of similarity.

In life we ​​meet not only with equal figures, but also with those that have the same shape, but different sizes. Geometry calls such figures similar.

All such figures have the same shape, but different sizes.

Definition: Two triangles are said to be similar if their angles are respectively equal and the sides of one triangle are proportional to the similar sides of the other.

If triangle ABC is similar to triangle A 1 B 1 C 1 , then the angles A, B and C are equal, respectively, to the angles A 1 , B 1 and C 1 ,
. number k, equal to the ratio similar sides of similar triangles is called the similarity coefficient.

Note 1: Equal Triangles are similar with a factor of 1.

Note 2: When designating similar triangles, their vertices should be ordered in such a way that the angles at them are equal in pairs.

Note 3: The requirements listed in the definition of similar triangles are redundant.

Properties of Similar Triangles

The ratio of the corresponding linear elements of similar triangles is equal to the coefficient of their similarity. Such elements of similar triangles include those that are measured in units of length. This is, for example, the side of a triangle, perimeter, median. Angle or area are not such elements.

The ratio of the areas of similar triangles is equal to the square of their similarity coefficient.

Signs of similarity of triangles .

If two angles of one triangle are respectively equal to two angles of another, then such triangles are similar.

If two sides of one triangle are proportional to two sides of another triangle and the angles included between these sides are equal, then such triangles are similar.

If three sides of one triangle are proportional to three sides of another triangle, then such triangles are similar.

1.3. Methods for measuring objects using similarity signs

1.3.1. First way measuring the height of an object

On a sunny day, it is not difficult to measure the height of an object, say a tree, by its shadow. It is only necessary to take an object (for example, a stick) of known length and set it perpendicular to the surface. Then a shadow will fall from the object. Knowing the height of the stick, the length of the shadow from the stick, the length of the shadow from the object whose height we measure, we can determine the height of the object. To do this, it is tedious to consider the similarity of two triangles. Remember: the sun's rays fall parallel to each other.

Parable

“A tired stranger came to the country of the Great Hapi. The sun was already setting when he approached the pharaoh's magnificent palace. He said something to the servants. In a moment the doors were opened for him and he was led into the reception hall. And here he stands in a dusty marching cloak, and in front of him sits a pharaoh on a gilded throne. Arrogant priests stand nearby, keepers of the great secrets of nature.

TO then you? the high priest asked.

My name is Thales. I am from Miletus.

The priest continued arrogantly:

So it was you who boasted that you could measure the height of the pyramid without climbing it? The priests doubled over with laughter. - It will be good, - the priest continued mockingly, - if you are wrong by no more than 100 cubits.

I can measure the height of the pyramid and be wrong by no more than half a cubit. I will do it tomorrow.

The faces of the priests darkened. What audacity! This foreigner claims to be able to figure out what they, the priests of great Egypt, cannot.

Good, said the pharaoh. There is a pyramid near the palace, we know its height. We'll check your art tomorrow."

The next day, Thales found a long stick, stuck it into the ground a little further away from the pyramid. Waiting for a certain moment. He made some measurements, said a method for determining the height of the pyramid and named its height. What did Thales say?



Thales's words : When the shadow from the stick has become the same length as the stick itself, then the length of the shadow from the center of the base of the pyramid to its top has the same length as the pyramid itself.

1.3.2.Second method measuring the height of an objectwas substantively described by Jules Verne in the novel "The Mysterious Island". This method can be used when there is no sun and no shadow from objects is visible. To measure, you need to take a pole equal in length to your height. This pole must be installed at such a distance from the object that, lying down, you can see the top of the object in one straight line with the top point of the pole. Then the height of an object can be found by knowing the length of the line drawn from your head to the base of the object.


An excerpt from a novel.

“Today we need to measure the height of the Far Cliff,” said the engineer.

Do you need a tool for this? Herbert asked.

No, it won't. We will act a little differently, turning to an equally simple and accurate method. The young man, trying to learn, perhaps more, followed the engineer, who descended from the granite wall to the edge of the coast.

Taking a straight pole, 12 feet long, the engineer measured it as accurately as possible, comparing it with his height, which was well known to him. Herbert carried behind him a plumb line handed to him by an engineer: just a stone tied to the end of a rope. Not reaching 500 feet from the granite wall, which rose sheer, the engineer stuck a pole about two feet into the sand and, having firmly strengthened it, set it vertically with a plumb line. Then he moved away from the pole at such a distance that, lying on the sand, one could see both the end of the pole and the edge of the ridge on one straight line. This point he carefully marked with a peg. Both distances were measured. The distance from the peg to the stick was 15 feet, and from the stick to the rock, 500 feet.

“-Do you know the rudiments of geometry? he asked Herbert, rising from the ground. Do you remember the properties of similar triangles?

-Yes.

-Their sides are proportional.

-Right. So: now I will build 2 similar right triangles. The smaller one has one leg, there will be a sheer pole, the other - the distance from the peg to the base of the pole; the hypotenuse is my line of sight. Another triangle will have legs: sheer wall, whose height we want to determine, and the distance from the peg to the base of this wall; the hypotenuse is my line of sight, coinciding with the direction of the hypotenuse of the first triangle. ... If we measure two distances: the distance from the peg to the base of the pole and the distance from the peg to the base of the wall, then, knowing the height of the pole, we can calculate the fourth, unknown term of the proportion, i.e. the height of the wall. Both horizontal distances were measured: the smaller one was 15 feet, the larger one was 500 feet. At the end of the measurements, the engineer made the following entry:

15:500 = 10:x; 500 x 10 = 5000; 5000: 15 = 333.3.

So the height of the granite wall was 333 feet.

1.3.3 Third method

Determining the height of an object using a mirror.

The mirror is placed horizontally and they move back from it to a point where, standing at which, the observer sees the top of the tree in the mirror. The light beam FD, reflected from the mirror at point D, enters the human eye. The measured object, for example a tree, will be as many times taller than you, in how much the distance from it to the mirror is greater than the distance from the mirror to you. Remember: the angle of incidence is equal to the angle of reflection (law of reflection).

AB D similar EFD (two corners) :

VA D = FED =90°;

    BUT D B = EDF , because the angle of incidence is equal to the angle of reflection.

In similar triangles, similar sides are proportional:



Chapter 2

2. 1. Measuring the height of an object

Let's take a tree as a measured object.

2.1.1. By the length of the shadow

This method is based on a modified method of Thales, which allows you to use a shadow of any length. To measure the height of a tree, it is necessary to stick a pole into the ground at some distance from the tree.

AB- tree height

BC- tree shadow length

A 1 B 1 - pole height

B 1 C 1 - the length of the shadow of the pole

B = < B 1 because the tree and the pole are perpendicular to the ground.

< A = < A 1 since the rays of the sun falling on the earth, we can consider parallel, because the angle between them is extremely small, almost imperceptible =>

Triangle ABC is similar to triangle A 1 In 1 With 1 .

After taking the necessary measurements, we can find the height of the tree.

AB= Sun.

A 1 B 1 B 1 C 1

AB = BUT 1 IN 1 ∙ Sun.

B 1 From 1

2.1.2 With a pole

A pole approximately equal to the height of a person is stuck into the ground vertically. The place for the pole must be chosen so that the person lying on the ground sees the top of the tree on the same line as the top of the pole.

ADE because< B = < D(corresponding),< A– general =>

AD = ED , ED =AD∙BC .

ABBCAB

ABOUT

A

B

C

A 1

C 1

shade height.


A 1 B 1 =1.6 m

BUT 1 FROM 1 =2.8 m

AC=17 m

2.1.3. With the help of a mirror.

At some distance from the tree, a mirror is placed on flat ground, and they move back from it to a point where the observer, standing, sees the top of the tree.

AB - tree height

AC - distance from the tree to the mirror

CD- the distance from the person to the mirror

ED- man's height.

Triangle ABC is similar to triangleDEC because

< A = < D(perpendicular)

< BCA = < ECD(since, according to the law of light reflection, the angle of incidence is equal to the angle of reflection.)

AC = AB ,

DC ED

AB=AC ∙ ED.

ABOUT
determination of the height of an object using a mirror.

AB=1.5 m

DE=12.5 m

AD= 2.7 m

2.1.4. What did the sergeant do?

Some of the methods of height measurement just described are inconvenient in that they make it necessary to lie down on the ground. Of course, this inconvenience can be avoided.

This is how it once happened on one of the fronts of the Great Patriotic War. Lieutenant Ivanyuk's division was ordered to build a bridge across a mountain river. The fascists settled on the opposite bank. For reconnaissance of the bridge construction site, the lieutenant assigned a reconnaissance group led by a senior sergeant. In a nearby wooded area, they measured the diameter and height of the most typical trees that could be used for building.

The height of the trees was determined using a pole as shown in Fig.

This method is as follows.

Stocking up on a pole taller than your height, stick it into the ground vertically at some distance from the tree to be measured. Step back from the pole, continueDd to that place A, from which, looking at the top of the tree, you will see on the same line with it the top pointbpole. Then, without changing the position of the head, look in the direction of the horizontal line aC, noticing the points c and C at which the line of sight meets the pole and the trunk. Ask an assistant to make notes in these places, and the observation is over.

< C = < cbecause the tree and the pole are perpendicular

< B = < bbecause the angle at which a person looks at a tree and at a pole is the same => triangleabcsimilar to a triangleaBC

=> BC = aC , BC = bc ∙aC .

bcacac

Distance bc, aCand ac is easy to measure directly. To the obtained value of the sun, you need to add the distanceCD(which is also measured directly) to find the desired height of the tree.

2.1.5 . Don't go near the tree.

It happens that for some reason it is inconvenient to come close to the base of the measured tree. Is it possible to determine its height in this case?

Quite possible. For this, an ingenious device was invented, which is easy to make yourself. two slatsad and with dfastened at right angles so thatab was equal to bc, but bdwas halfad. That's the whole device. To measure their height, hold it in their hands, opposite the barcdvertically (for which it has a plumb line - a string with a weight), and becomes sequentially in two places: first at point A, where the device is placed with the end up, and then at point A`, further away, where the device is held upside downd. Point A is chosen so that, looking from a to end c, we see it on the same line as the top of the tree. Point

A` is found in such a way that, looking from a` at the pointd`, to see it coinciding with V.

The triangle BCA is similar to the trianglebca because

< C = < b(perpendicular)

< B = < c(observer looks at one angle)

The triangle BCa` is similar to the triangleb` d` a` because

< C = < b` (perpendicular)

< B = < d` (observer looks at one angle)

The whole measurement consists in finding two points A and A`, because the desired part of BC is equal to the distance AA`. Equality follows from the fact that aC \u003d BC, since the triangleabcisosceles (by construction). Hence the triangleaBCisosceles. a`C = 2 BCfollows from the relations in similar triangles; means,a` C – aC = BC.

ABOUT
height measurement using a right-angled isosceles triangle.

CD = AB + BD

AB = 8.9 m

BD =1.2 m

FROM D =8.9+1.2≈10 m

2.2. Pond cleaning.

There is a pond in the village of Kirov, which is very polluted. We decided to find out how to clean it.

2.2.1. Methods for cleaning water bodies.

Cleaning of reservoirs is carried out by mechanized, hydromechanized, explosive and manual methods. The most common of all methods is mechanical. With this method, cleaning with a dredger is used.

Dredger NSS - 400/20 - GRProductivity (soil alluvium): 800m/cube per shift. Dimensions: length 10 m, width 2.7 m, height 3.0 m.Weight: 17 tons. Slurry pipeline: 100m (including 50m floating, 50m onshore). The dredge is equipped with an arrow. Boom length - 10 m, with hydraulic washout (supply of 60 m3/cube per hour of water at a head of 40 m, pump power 7 kW).Engine: D-260-4. 01 (210 l/s, fuel consumption - 14 l/h, speed - 1800 rpm). Pump: GRAU 400/20. Technical characteristics of the pump: soil output 10-30% per hour, water column head - 20m, max power - 75 kW, rotational speed - 950 rpm. The dredger of this modification raises the soil from the depth of the reservoir 1-9.5 m, pushing through the slurry pipeline up to 200 m. Pipeline diameter: 160 mm. Energy supply: autonomous. Movement with winches - 4 engines of 1.5 kW.

In our particular case, we are interested in the length of the dredger boom - 10m.

2.2.2.Measuring the width of the pond.

The properties of such triangles can be used to carry out various measurements on the ground. We will consider one problem: determining the distance to an inaccessible point. For example, we will try to measure the width of a pond using triangle similarity signs.

So, with the help of some instruments and calculations, let's get to work. To get more accurate results, we measured the pond in two places.

Suppose we need to find the distance from point A on the shore on which we are standing to the pointBlocated on the opposite bank of the river. To do this, we select point C on “our” shore, simultaneously measuring the resulting segment AC. Then, using the astrolabe, we measure angles A and C. On a piece of paper we build a triangle A 1 B 1 C 1 , so that 1 sign of similarity of triangles is observed (at 2 corners). Injection A 1 is equal to angle A, and the angleC 1 equal to the angleC. We measure the sides A 1 B 1 And A 1 C 1 triangle A 1 B 1 C 1 .Because the trianglesABCAnd A 1 B 1 C 1 are similar, thenAB/ A 1 B 1 = AC/ A 1 C 1 , whence we getAB = AC* A 1 B 1 / A 1 C 1 This formula allows for known distancesAC, A 1 C 1 And A 1 B 1 find distanceAB.

Devices:

Astrolabe, demonstration ruler (or, for example, a rope about 4 m long).

Preliminary measurements:

We have measured the pond in two places, so we will describe each measurement in turn.

1) Let's take any point on the opposite bank, located near the border of the pond and the earth, say, a small hole or, if we prepare in advance, a peg driven into the ground, a milestone.


It turned out 88 degrees, we have the first corner. In the same way, placing the device on point C, located at a distance, in our case, 4 meters from point A, we measure the angle C. 70 degrees. And, in fact, the measurements ended there.

2) At the second place, where we measured the width of the river, we got approximately equal angles with the first case: A=90, C=70 degrees.


Calculations:

    Draw a triangleA 1 B 1 C 1 , in which the angle A 1 =88 , and the angleC 1 =70 degrees. SectionA 1 C 1 , for ease of measurement, we take equal to 4 centimeters. Now we measure the segmentA 1 B 1 . It turned out about 11 cm. We translate the results into meters and collect them in proportion:

AB/A 1 B 1 =AC/A 1 C 1

AB-? ;A 1 B 1 =0,11 m; AC=4m; A 1 C 1 =0,04 m.

We expressAB:

AB=AC*A 1 B 1 / A 1 C 1 ;

AB=4*0,11/0,04;

AB=0.44/0.04=11m

So, in the first case, the width of the pond is 11 m.

    Following the same method, we find all the sides and make up the proportion. But the results, since the angles are approximately equal, are the same. So, we measured the width of the pond in two places and got one result - 11 meters.

Earlier, I indicated that the length of the dredger boom is 10 meters, i.e. it is quite enough to clean the pond from one side.

So, my assumption is that geometry, and in this case the similarity of triangles, helps to solve social problems right. I proved that with the help of similarity it is possible to calculate the height of buildings and the width of a pond.

After all, sometimes you really want your native corner, the place in which we live, to shine with new colors, to cause pride. I want to go down anywhere to a river or a pond and swim without fear for my health. I want to be proud of my small Motherland. And for this we must all try. Everything is in our hands.

I explored various ways to measure the height and width of objects in the area using similar triangles.

Conclusion

I learned a lot about applying similar triangles.

How to find the distance to an inaccessible point? How, by constructing similar triangles, to find the distance between two inaccessible points A and B? How to find the height of an object whose base can be approached?

The solution of such problems contributes to the development of logical thinking, the ability to analyze the situation, and the use of the triangle similarity method in solving them, thereby increases the mathematical culture, developing mathematical abilities.You can use the geometric material I have considered both in the lessons of geometry and physics, and in preparation for the State Final Attestation,

Geometry is a science that has all the properties of crystal glass, is just as transparent in reasoning, impeccable in evidence, clear in answers, harmoniously combining the transparency of thought and the beauty of the human mind. Geometry is not a fully understood science, and maybe many discoveries are waiting for you.

Literature:

1. Glazer G.I. History of mathematics at school 7-8 cells. - M.: Enlightenment, 1982.-240 p.

2. Savin A.P. I know the world - M .: AST-LTD Publishing House LLC, 1998.-480 p.

3. Savin A.P. encyclopedic Dictionary young mathematician. - M.: Pedagogy, 1989, -352 p.

4. Atanasyan L.S. etc. Geometry 7-9: Proc. for general education institutions. - M.: Enlightenment, 2005, -245s.

5. G.I. Bavrin. Great student guide. Maths. M. bustard. 2006 435s

6.I. I. Perelman. Interesting geometry. Domodedovo. 1994 11-27s.

7. http:// canegor. urc. ac. en/ zg/59825123. html

The work was based on the study of the possibility of using the similarity of triangles in real life, experiments were carried out on measuring the length using an altimeter.


"11Sushko-t.doc"

SIMILARITY OF TRIANGLES IN REAL LIFE

Sushko Daria Olegovna

8th grade student

KU "OSHI - III steps No. 11, Enakievo "

Ikaeva Marina Alexandrovna

Mathematic teacher,II category

KU "OSHI - III steps No. 11, Enakievo "

[email protected]

Geometry originated in ancient times. The world we live in today is also filled with geometry. All objects around us are geometric shapes. These are buildings, streets, plants, household items. The relevance of my topic lies in the fact that without any tools, only relying on the similarity of triangles, you can measure the height of a pillar, bell tower, tree, the width of a river, lake, ravine, the length of an island, the depth of a pond, etc.

The aim of the work was to find areas of application of the similarity of triangles in real life.

The tasks of the work were

Objects and subjects of research : height: post; tree, pyramid model.

In the course of the work, the following methods were applied: literature review, practical work, comparison.

The work is practice-oriented practical significance work lies in the possibility of using the results of the study in geometry lessons, in Everyday life.

As a result of the work, measurements of the height of the pillar, tree, models made by the author were carried out.

View document content

Content:

    Introduction

    The concept of similarity of figures. Signs of similarity.

4.1 Determining the height from the shadow

4.2. Jules Verne Height Measurement

4.3. Altitude measurement with an altimeter

5. Conclusions

    Introduction.

Geometry originated in ancient times. Building dwellings and temples, decorating them with ornaments, marking the ground, measuring distances and areas, a person applied his knowledge of the shape, size and relative position of objects, obtained from observations and experiments. The world we live in today is also filled with geometry. All objects around us have geometric shapes. These are buildings, streets, plants, household items. In everyday life, there are often figures of the same shape, but of different sizes. Such figures in geometry are called similar. My work is devoted to the similarity of triangles, because, studying this topic in mathematics lessons, I was interested in how the concept of triangle similarity and similarity signs are applied in practice. The relevance of my topic lies in the fact that without any tools, you can measure the height of a pillar, a bell tower, a tree, the width of a river, lake, ravine, the length of an island, the depth of a pond, etc.

The tasks of my work were

    study the literature on the topic;

    study the history of the concept of similarity;

    find out where the similarity of triangles is used;

    measure the height of the pillar using the similarity of triangles in various ways;

2. The legend about Thales measuring the height of the pyramid.

Much is associated with the pyramid. mysterious stories and legends. On one of the hot days, Thales, along with the chief priest of the temple of Isis, walked past the pyramid of Cheops.

Look, - Thales continued, - just at this time, no matter what object we take, the shadow from it, if you put it vertically, exactly the height of the object! In order to use the shadow to solve the problem of the height of the pyramid, it was necessary to already know some geometric properties of the triangle, namely the following two (of which Thales discovered the first himself):

1. That the angles at the base of an isosceles triangle are equal, and vice versa - that the sides lying opposite the equal angles of the triangle are equal to each other; 2. That the sum of the angles of any triangle is equal to two right angles.

Only Thales, armed with this knowledge, had the right to conclude that when his own shadow is equal to his height, the sun's rays meet even ground at an angle of half a right, and therefore, the top of the pyramid, the middle of its base and the end of its shadow should indicate an isosceles triangle. This in a simple way It would seem that it is very convenient to use on a clear sunny day to measure lonely standing trees, the shadow of which does not merge with the shadow of neighboring ones. But in our latitudes it is not as easy as in Egypt to lie in wait for the right moment for this: our Sun is low above the horizon, and the shadows are equal to the height of the objects casting them only in the near noon hours of the summer months. Therefore, the method of Thales in the specified form is not always applicable.

The doctrine of the similarity of figures based on the theory of relations and proportions was created in Ancient Greece in the V-IV centuries. BC e. It is set out in the VI book of Euclid's "Beginnings" (III century BC), beginning with the following definition: "Similar rectilinear figures are those that have respectively equal angles and proportional sides"

3. The concept of similar figures.

In life, we meet not only with equal figures, but also with those that have the same shape, but different sizes. Geometry calls such figures similar. Similar triangles are triangles in which the angles are respectively equal, and the sides of one are proportional to the similar sides of the other triangle. Similarity criteria for triangles are geometric features that allow you to establish that two triangles are similar without using all the elements.

Signs of similarity of triangles.

4. Measurement work with the help of similarity.

4.1. Determination of height by shadow.

I decided to conduct an experiment to determine the height of the shadow.

For this I needed: a flashlight, a pyramid layout, a figurine. Making a miniature pyramid for experiments is easy. I needed: a sheet of paper; pencil; ruler; scissors; paper glue. On a sheet of paper, I built a pyramid scan, at the base of which is a square with a side of 7.6 cm, and tank faces are equal isosceles triangles with a side of 9.6 cm. The height of the resulting pyramid is 7.9 cm. The height of the figurine is 8.1 cm. Let's try to measure the height of this pyramid by its shadow, using also the shadow of the figurine. On a sunny day, I measured the shadow of the pyramid and figurines. I got: 15 cm - the shadow of the figure, 13 cm - the shadow of the pyramid.

Let us construct a geometric model of this problem:

, ∠ ACO= ∠ MLK as the angles of incidence of the sun's rays, which means at two angles.

Let us now find the height of the pyramid in another way to compare the results. Find the height of the side face: AB =

From we find the height AO \u003d

We got almost the same results. Having received such results, I decided to measure the height of the pole, going outside.

I chose a pillar that cast a clear shadow and measured it. It was 21 m. Then I stood next to the pole and my assistant measured my shadow, it was 4.5 meters. My height, given that I was in shoes and a headdress, was 1.6.

Let's find the height of the column by compiling a geometric model of the problem.

Consider , KO - the length of my shadow, BC - the length of the shadow of the column. AB - desired.

∠ABC=∠CIE= as the angles of incidence of the sun's rays.

4.2. Measuring the height of the pyramid by the Jules Verne method.

The Mysterious Island describes an interesting way of determining the height: “The young man, trying to learn as much as possible, followed the engineer, who descended from the granite wall to the edge of the coast. Taking a straight pole, 12 feet long, the engineer measured it as accurately as possible, comparing it with his height, which was well known to him. Herbert carried behind him the plumb line handed to him by the engineer: just a stone tied to the end of a rope. Not reaching 500 feet from the granite wall, which rose sheer, the engineer stuck a pole two feet into the sand and, firmly strengthening it, set it vertically with a plumb line. Then he moved away from the pole at such a distance that, lying on the sand, it was possible to lines to see both the end of the pole and the edge of the ridge. this point he carefully marked with a peg.

Do you know the basics of geometry? he asked Herbert, rising from the ground.

Do you remember the properties of similar triangles?

Their corresponding sides are proportional. - Right. So: now I will build two similar right triangles. The smaller one will have a vertical pole with one leg, the distance from the peg to the base of the pole will be the other; the hypotenuse is my line of sight. In another triangle, the legs will be: a sheer wall, the height of which we want to determine, and the distance from the peg to the base of this wall; the hypotenuse is the line of sight, coinciding with the direction of the hypotenuse of the first triangle.

Understood! - exclaimed the young man. - The distance from the peg to the pole is related to the distance from the peg to the base of the wall, as the height of the pole is to the height of the wall. - Yes. And consequently, if we measure the first two distances, then, knowing the height of the pole, we can calculate the fourth, unknown term of the proportion, i.e., the height of the wall. We will, therefore, dispense with the direct measurement of this height. Both horizontal distances were measured: the smaller one was 15 feet, the larger one was 500 feet. At the end of the measurements, the engineer made the following entry:

4.3 Determining altitude with an altimeter

Height can be measured with a special device - an altimeter. For the manufacture of this device you will need: Thick white cardboard, ruler, pen, pencil, scissors, thread, weight, needle.

7. On it, we bend two rectangles 3x5 cm in size from the sides and cut two holes with different diameters: one is smaller - near the eye, the other is larger - in order to point to the top of the tree. So, I decided to conduct an experiment and test this method of measuring the height of an object. As a measured object, I chose a tree growing near the school.

I moved away from the measured object by 21 steps, that is, EO = 6.3 m. I measured the readings of the device, it showed 0.7. My height is 1.6 m. It is required to find the height of the tree.

To do this, we construct a geometric model of this problem:

=

Let's add my height to the obtained value and get: LV \u003d LO + OV \u003d 3.71

1.6=5.31 is the height of the tree.

Also, I could make mistakes in using the device. Errors in the use and manufacture of the device:

1. If you do not bend the upper rectangle from the base, then you will incorrectly determine the height.

2. When measuring the height of an object, the weight must be directed to a specific markup value.

3. The distance from the measured object must be accurate.

4. Accurately apply 1 cm markings.

The experiment showed that the method of determining the height of an object using the "altimeter" device is more accurate and convenient.

5. Conclusions.

Literature

5. Perelman Ya. I. Entertaining geometry. - M .: State publishing house of technical and theoretical literature, 1950
There are 3 ways to measure the height of a tree.

1. General dictionary Russian language [ Electronic resource]. – Access mode: http://tolkslovar.ru/p22702.html

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Communal institution " Comprehensive school I-III stages No. 11, Enakievo "

"Mathematics around us"

Creative work on the theme

"The similarity of triangles in real life"

Performed

8th grade student

Sushko Daria

Supervisor

mathematic teacher

Ikaeva Marina Alexandrovna

Enakievo 2017

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"The similarity of triangles in real life"


KU "Secondary school of І-ІІІ stages No. 11 of Enakievo"

Competition of student creative projects

"Mathematics around us"

Creative work on the theme

"The similarity of triangles in real life"

Performed

8th grade student

Sushko Daria

Supervisor

mathematic teacher

Ikaeva Marina Alexandrovna

Enakievo 2017


The purpose of my work was to find applications for the similarity of triangles in real life.

The tasks of my work were

  • study the literature on the topic;
  • study the history of the concept of similarity;
  • find out where the similarity of triangles is used;
  • measure the height of the pillar using the similarity of triangles in various ways;

The legend of Thales measuring the height of the pyramid

On one of the hot days, Thales, along with the chief priest of the temple of Isis, walked past the pyramid of Cheops.

Does anyone know what its height is? - he asked.

No, my son, - the priest answered him, - the ancient papyri did not preserve this for us. “But you can determine the height of the pyramid quite accurately and right now!” Thales exclaimed.

Look, - Thales continued, - just at this time, no matter what object we take, the shadow from it, if you put it vertically, exactly the height of the object!


concept similarities figures

Similar triangles are triangles in which the angles are respectively equal, and the sides of one are proportional to the similar sides of the other triangle.

Two figures are called similar if they are converted into each other by a similarity transformation.

Similarity criteria for triangles are geometric features that allow you to establish that two triangles are similar without using all the elements.

If two angles of one triangle are respectively equal to two angles of another, then such triangles are similar.

If two sides of one triangle are proportional to two sides of another triangle and the angles included between these sides are equal, then such triangles are similar.

If three sides of one triangle are proportional to three sides of another triangle, then such triangles are similar.


Shadow Height Measurement

The initial data of the problem: The length of the shadow of the pyramid BC = 11 cm, the length of the shadow of the figure KL = 15 cm, the height of the figure KM = 8 cm, the base of the pyramid is a square with a side of 7.6 cm. The height of the pyramid AO is the required one.

Consider right triangles AOS and MKL:

, ∠ ACO= ∠ MLK as the angles of incidence of the sun's rays, which means in two angles.


Measuring the height of a pillar by its shadow

Consider, KO is the length of my shadow, BC is the length of the shadow of the pillar. AB - desired.

∠ ABC=∠CIE= as the angles of incidence of the sun's rays.

Thus, I got an approximate value for the height of the column 7.46 m.


Jules Verne Height Measurement

This method consists in the fact that you need to drive a pole into the ground, lie down on the ground so that you can see the upper end of the pole and the top of the object being measured. Measure the distance from the pole to the object, measure the height of the pole and the distance from the top of the person to the base of the pole.

In Jules Verne's novel The Mysterious Island Both horizontal distances were measured: the smaller one was 15 feet, the larger one was 500 feet. At the end of the measurements, the engineer made the following entry:

15: 500 = 10:x, 500 X 10 = 5000, 5000: 15 = 333.3.


Altitude measurement with an altimeter

1. We draw and cut out a 15x15cm square from cardboard.

2. Divide the square into two rectangles: 5x15 cm, 10x15 cm.

3. We divide the rectangle 10x15 cm into two parts: 5 cm and 10 cm.

4. For the most part with a length of 10 cm, we apply centimeter divisions and designate them decimal, that is, 0.1; 0.2, etc.

5. At point E, make a hole with a needle and pull the thread with a weight, and then fasten the thread at the back.

6. In order to make it more convenient to look at, we bend the upper rectangle from the base.

7. On it, we bend two rectangles 3x5 cm in size from the sides and cut two holes with different diameters: one is smaller - near the eye, the other is larger - in order to point to the top of the tree.


Altitude measurement with an altimeter

To find the height of the LV, you need to add your height to the LO.

LV \u003d LO + OV \u003d 3.71 + 1.6 \u003d 5.31 - the height of the tree.


Conclusions:

Having completed my work, I learned that there are many various ways determining the height of an object. I conducted an experiment to determine the height of an object by its shadow. I did the test at home on the model of the pyramid and figurine, as well as on the street when measuring the height of the pillar. Also, I looked at the Jules Verne method for determining height. I studied the concept of an altimeter and made an altimeter device, which I put into practice to measure the height of a selected object. The most convenient way to measure altitude for me was to use an altimeter. Thus, the objectives of my work have been achieved. We can safely say that the similarity of triangles is used in real life for measuring work on the ground.


Literature:

1. Glazer G.I. History of mathematics at school. - M .: Publishing house "Enlightenment", 1964.

2. Perelman Ya. I. Entertaining geometry. - M .: State publishing house of technical and theoretical literature, 1950.

3.J.Vern. Mysterious Island. - M: Children's Literature Publishing House, 1980.

4. Geometry, 7 - 9: textbook. for general education institutions / L.S. Atanasyan, V.F. Butuzov, S.B. Kadomtsev and others - 18th ed. - M.: Education, 2010 Used materials and Internet resources.

5. Perelman Ya. I. Entertaining geometry. - M .: State publishing house of technical and theoretical literature, 1950 You can measure the height of a tree in 3 ways.

1. General explanatory dictionary of the Russian language [Electronic resource]. - Access mode: http://tolkslovar.ru/p22702.html

2. Figure 2 [Electronic resource]. – Access mode: http://www.dopinfo.ru


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