Finishing 

Message on the topic of measuring work on the ground. Measuring work. Criteria for assessing the achievement of expected results

During excursions, hikes or work on an expedition, it often becomes necessary to measure the distance between objects, sometimes a small area, or even height, make a profile along the route, etc. There are many ways to measure distances, angles, elevations and heights on the ground. Let's get acquainted with the simplest of them.

Distance can be measured in steps. In adults, the step is on average 0.7-0.8 m. Two steps are taken as 1.5 m. When measuring distances, steps are considered in pairs. Long distances are measured by the time spent walking. average speed human movement with a normal step - 5 km / h. For greater measurement accuracy, this method carefully determines the speed of movement. With high precision small distances are measured with a tape measure or a steel measuring tape, the length of which is usually 20 m. The "two-meter" has found wide application in agriculture. With this method, 1 m of error is possible for every hundred meters.

Angular measurements are used in orientation, in determining the location of various objects, the direction of movement. To measure angles, make a goniometer. Fold a piece of cardboard into a square folder. From the top of the corner, draw an arc with a radius equal to the side of the square. With the same radius, lay a chord on the arc. Its ends will limit the arc of a circle with a central angle of 60 °. Divide the chord into 6 equal parts and one right side into another 10 equal parts. Each large division will correspond to 10°, and each small division to 1°. At the division points on the top cover, poke holes for the pins through which they must pass and stick into the bottom cover. Attaching the eye, as shown in the figure, determine the direction of the line of sight on one object by the inserted pin, and on the other object, also insert a pin on this line. Count the number of tens and units of degrees between the pins. In the example shown in the figure, the angle value is 34°.

Determining the excess of some points of the terrain over others is called leveling. Let's make a homemade level. Two planks: one is 1 m long, the second is 1.5 m. To the end of the first, we will nail a small rectangular piece of plywood. We fix the thread with a weight at its top, and the level is ready (see fig.). From the second bar we will make a leveling rail. We mark a meter segment on it and divide it into 10 equal parts, 10 cm each. The countdown can be taken by eye up to 0.1 parts, that is, with an accuracy of 1 cm. The divisions on the rail are signed from the middle of the meter section up and down, as it shown on the picture. Excesses are determined as follows: a level is placed on one point, a rail on the second. They sight along the level installed on a plumb line and make a reading along the rail. In our drawing, it is 23 cm. This means that the excess of one point over the other is 23 cm. Proof: the zero mark on the rail is at the same distance from the surface of the earth as the upper side of the level.

The simplest way to determine the relative height of objects is with a protractor made from a school right-angled isosceles triangle. A ruler D is nailed to it (see p. 158) and divides one of the acute angles of the triangle so that the angle VBG is 22 °. A plumb line is strengthened on side AB in such a way that its end coincides with the index G at the end of the ruler D. To determine the height of an object, they move away from it at a distance from which you can sight at its top along the hypotenuse AB with the leg AB vertical. This can only be done from a point separated from the object at a distance equal to its height (see Fig.). Therefore, sighting point C is at a distance of CE = ET from the measured outcrop. The exposure height is equal to ET+h, where h is the excess of the device above the earth's surface.

If it is impossible to approach the object being measured, its height is measured as shown in the figure. First, they retreat to a distance from which it is possible to sight on its top along the hypotenuse of the goniometer. This can be done from point C. Then they move away from this point to such a distance from which the same exposure height can be sighted along the ruler on the goniometer. This can only be done from point 3. If you measure the distance of the AP, then it will be equal to the height ET. To it must be added the height of the observer's eye above the earth's surface (h).

Often there are more complex jobs to be done. For example, schoolchildren decided to help the collective farm study the relief of a plot chosen for a garden or for building a house, for a road, canal, etc. For the correct organization of work, it is necessary to know the relief of the site. For this purpose, leveling is carried out, i.e., the difference in heights of various points, their excesses are determined. They will characterize the relief. First, points are marked and the distance between them is measured. Then leveling is performed, the excess between the points is calculated. This excess has a plus or minus sign. For a visual representation of the relief, a drawing is built - a profile between points, on which the relief is depicted. To do this, a horizontal line is drawn, on which the distances between points are plotted on a certain scale. At the points obtained, perpendiculars are built and the heights of the points are laid on them on a different scale. For example, in the figure, the profile is built on a scale of 1:1000 horizontally and 1:100 vertically. It is more convenient to build a profile on graph paper or checkered paper. After connecting the points of heights, a broken line is obtained, depicting a vertical section earth's surface. If the relief is being studied in a certain area (and not a track), then a series of profiles is built in different directions.

When we determined the heights of points, the relative height was found out, that is, the excess of one point on the earth's surface relative to another point, in other words, the difference in the absolute heights of these points. Absolute height, or absolute mark, is the vertical distance of any point on the surface of the earth from the average level of the ocean surface. In the USSR, the absolute height is measured from the level Baltic Sea, for which the zero of the footstock (water gauge) in Kronstadt is taken. The absolute height of points above this level is positive, below it is negative. It is determined by leveling from a point whose absolute height is known, for example, shown on a topographic map.

Much more complex measuring work is carried out by specialists in topographic surveys or geodetic measurements. For this purpose, it is necessary to build a network of strong points on the ground, consisting of a system of triangles in which the angles are measured, and in the network, the length of at least one side (basis); from trigonometric calculations find the relative position of all points. The determined points serve as the vertices of triangles, which are marked on the ground with signs installed on elevated places; they are at a distance of several kilometers from each other, but in such a way that there is mutual visibility between adjacent signs. This method of determining the position of geodetic points is called triangulation (see Geodesy).

In the course of studying the geometry of the main school, tasks related to the practical application of the studied knowledge are considered: measuring work on the ground, measuring tools. Practical work on the ground is one of the most active forms of linking learning with life, theory with practice. Students learn to use reference books, apply the necessary formulas, master the practical techniques of geometric measurements and constructions.

Practical work using measuring instruments increases students' interest in mathematics, and solving problems on measuring the width of a river, the height of an object and determining the distance to an inaccessible point allow them to be applied in practice, to see the scale of the application of mathematics in human life.

As the material is studied, the ways of solving these problems change, the same problem can be solved in many ways. In this case, the following questions of geometry are used: equality and similarity of triangles, relations in a right triangle, the sine theorem and the cosine theorem, the Pythagorean theorem, properties of right triangles, etc.

The objectives of the lessons "Measurement on the ground":

Tasks:

  • scientific character;
  • visibility;
  • differentiated approach;

Criteria for assessing the achievement of expected results:

  • student activity;

Preparing and conducting such lessons will result in:

  • to teach how to apply mathematical knowledge in everyday practical life.

One of the most active forms connection of learning with life, theory with practice is the fulfillment by students in geometry lessons practical work associated with measurement, construction, image. In the course of studying the geometry of the main school, tasks related to the practical application of the studied knowledge are considered: measuring work on the ground, measuring tools. In mathematics lessons, in parallel with the study of theoretical material, students must learn to make measurements, use reference books and tables, and be fluent in drawing and measuring tools. Work is carried out both on the ground and problem solving in the classroom different ways to find the height of an object and determine the distance to an inaccessible point. The following topics are covered in the geometry course:

7th grade

  • “Hanging a straight line on the ground” (p. 2),
  • “Measuring tools” (item 8),
  • “Measurement of angles on the ground” (p. 10),
  • “Construction of right angles on the ground” (p. 13),
  • “Problems for construction. Circle” (item 21),
  • “Practical Ways construction of parallel lines” (item 26),
  • “Corner reflector” (item 36),
  • “Distance between parallel lines” (item 37 - thickness gauge),
  • “Construction of a triangle by three elements” (item 38)

8th grade.

  • “Practical applications of the similarity of triangles” (p. 64 - determining the height of an object, determining the distance to an inaccessible point)

Grade 9

  • “Measuring work” (item 100 - measuring the height of an object, measuring the distance to an inaccessible point).

Practical work in geometry lessons allows solving pedagogical problems: posing a cognitive mathematical problem for students, updating their knowledge and preparing them to assimilate new material, forming practical skills in handling various devices, tools, computer technology, reference books and tables.. They allow you to implement in teaching the most important principles of the relationship between theory and practice: practice acts as the initial link in the development of theory and serves as the most important incentive for students to study it, it is a means of testing the theory and the area of ​​its application.

The system of conducting lessons "Measurement on the ground" sets the following goals:

  • practical application of theoretical knowledge of students;
  • activation of cognitive activity of students;

Provides for the following tasks:

  • expanding the horizons of students;
  • increased interest in the subject;
  • development of ingenuity, curiosity, logical and creative thinking;
  • the formation of the qualities of thinking that are characteristic of mathematical activity and necessary for a productive life in society.

When selecting the content of each lesson on a given topic and forms of student activity, the following principles are used:

  • relationship between theory and practice;
  • scientific character;
  • visibility;
  • taking into account the age and individual characteristics of students;
  • combinations of collective and individual activities of participants;
  • differentiated approach;

Criteria for assessing the achievement of expected results:

  • student activity;
  • independence of students in the performance of tasks;
  • practical applications mathematical knowledge;
  • the level of creativity of the participants.

Preparing and conducting such lessons will result in:

  • connect, awaken and develop the potential abilities of students;
  • identify the most active and capable participants;
  • to educate the moral qualities of a person: diligence, perseverance in achieving goals, responsibility and independence.
  • teach how to apply mathematical knowledge in everyday practical life;
  • handle various devices, tools, computers, reference books and tables.

Measuring tools used in field measurements:

  • Roulette - a tape with divisions applied to it, designed to measure distance on the ground.
  • Eker is a device for constructing right angles on the ground.
  • The astrolabe is a device for measuring angles on the ground.
  • Milestones (poles) - stakes that are driven into the ground.
  • Surveying compasses (field compasses - sazhen) - a tool in the form of the letter A 1.37 m high and 2 m wide. To measure the distance on the ground, it is more convenient for students to take the distance between the legs 1 meter.

Ecker

Eker consists of two bars located at a right angle and mounted on a tripod. At the ends of the bars, nails are driven in so that the straight lines passing through them are mutually perpendicular.

Astrolabe

Device: the astrolabe consists of two parts: a disk (limb), divided into degrees, and rotating around the center of the ruler (alidade). When measuring an angle on the ground, it is aimed at objects lying on its sides. Alidade guidance is called sighting. Diopters are used for sighting. These are metal plates with slots. There are two diopters: one with a cut in the form of a narrow slit, the other with a wide cut, in the middle of which a hair is stretched. When sighting, the eye of the observer is applied to a narrow slit, therefore a diopter with such a slit is called an eye diopter. A diopter with a hair is directed to an object lying on the side of the measured; it is called subject. A compass is attached to the middle of the alidade.

astrolabe

Practical work

1. Building a straight line on the ground (fixing a straight line)

Segments on the ground are indicated by milestones. In order for the pole to stand straight, a plumb line is used (some kind of weight suspended on a thread). A number of milestones driven into the ground and denotes a segment of a straight line on the ground. In the chosen direction, two milestones are placed at a distance from each other, there are other milestones between them, so that looking through one, the others are covered by each other.

Practical work: building a straight line on the ground.

Task: mark on it a segment of 20 m, 36 m, 42 m.

2. Measuring the average stride length.

A certain number of steps is counted (for example, 50), this distance is measured and the average step length is calculated. It is more convenient to carry out the experiment several times and calculate the arithmetic mean.

Practical work: measuring the average stride length.

Task: knowing the average step length, set aside a segment of 20 m on the ground, check with a tape measure.

3. Construction of right angles on the ground.

To build a right angle AOB with a given side OA on the ground, a tripod with an ecker is installed so that the plumb line is exactly above the point O, and the direction of one bar coincides with the direction of the beam OA. The combination of these directions can be done with the help of a milestone placed on the beam. Then hang a straight line in the direction of another bar (OB).

Practical work: building right angle on the ground, rectangle, square.

Task: measure the perimeter and area of ​​a rectangle, square.

4. Construction and measurement of angles using an astrolabe.

The astrolabe is installed at the top of the measuring angle so that its limb is located in a horizontal plane, and a plumb line suspended under the center of the limb is projected to a point taken as the top of the angle on the earth's surface. Then they sight with alidade in the direction of one side of the measured angle and count the degree divisions on the limb against the mark of the subject diopter. The alidade is rotated clockwise in the direction of the second side of the corner and a second count is taken. The desired angle is equal to the difference between the readings at the second and first readings.

Practical work:

  • measurement of given angles,
  • construction of angles of a given degree measure,
  • construction of a triangle according to three elements - along the side and two angles adjacent to it, along two sides and the angle between them.

Task: measure the degree measures of given angles.

5. Construction of a circle on the ground.

A peg is set on the ground, to which a rope is tied. Holding on to the free end of the rope, moving around the peg, you can describe a circle.

Practical work: construction of a circle.

Task: measurement of radius, diameter; calculation of the area of ​​a circle, the circumference of a circle.

6. Determining the height of an object.

a) With the help of a rotating bar.

Suppose that we need to determine the height of some object, for example, the height of a column A 1 C 1 (problem No. 579). To do this, we put a pole AC with a rotating bar at a certain distance from the pole and direct the bar to the top point C 1 of the pole. Let us mark a point B on the surface of the earth, at which the line A 1 A intersects with the surface of the earth. Right triangles A 1 C 1 B and DIA are similar in the first sign of similarity of triangles (angle A 1 \u003d angle A \u003d 90 o, angle B is common). From the similarity of triangles it follows;

By measuring the distances VA 1 and VA (the distance from point B to the base of the column and the distance to the pole with a rotating bar), knowing the length of the AC pole, using the resulting formula, we determine the height A 1 C 1 of the column.

b) With the help of a shadow.

The measurement should be taken in sunny weather. Let's measure the length of the shadow of a tree and the length of a person's shadow. Let's construct two right-angled triangles, they are similar. Using the similarity of triangles, we will compose a proportion (the ratio of the corresponding sides), from which we will find the height of the tree (problem No. 580). It is possible in this way to determine the height of the tree in 6 cells, using the construction of right-angled triangles on the selected scale.

c) Using a mirror.

To determine the height of an object, you can use a mirror located horizontally on the ground (problem No. 581). A beam of light reflected from a mirror hits a person's eye. Using the similarity of triangles, you can find the height of an object, knowing the height of a person (to the eyes), the distance from the eyes to the top of the person and measuring the distance from the person to the mirror, the distance from the mirror to the object (given that the angle of incidence of the beam is equal to the angle of reflection).

d) Using a drawing right triangle.

We place a right triangle at eye level, pointing one leg horizontally to the surface of the earth, and pointing the other leg at the object whose height we are measuring. We move away from the object at such a distance that the second leg “covers” the tree. If the triangle is also isosceles, then the height of the object is equal to the distance from the person to the base of the object (adding the height of the person). If the triangle is not isosceles, then the similarity of triangles is used again, measuring the legs of the triangle and the distance from the person to the object (the construction of right-angled triangles on the selected scale is also used). If the triangle has an angle of 30 0, then the property of a right-angled triangle is used: opposite the angle of 30 0 lies the leg half the hypotenuse.

e) During the game “Zarnitsa”, students are not allowed to use measuring instruments, so the following method can be suggested:

one lies on the ground and directs his eyes to the crown of the other, who is at a distance of his height from him, so that the straight line passes through the crown of his friend and the top of the object. Then the triangle turns out to be isosceles and the height of the object is equal to the distance from the object lying to the base, which is measured, knowing the average length of the student's step. If the triangle is not isosceles, then knowing the average step length, the distance from the one lying on the ground to the one standing and to the object, the growth of the one standing is known, is measured. And then, based on the similarity of triangles, the height of the object is calculated (or the construction of right-angled triangles in the selected scale).

7. Determining the distance to an inaccessible point.

a) Suppose we need to find the distance from point A to inaccessible point B. To do this, select point C on the ground, hang the segment AC and measure it. Then, using an astrolabe, we measure angles A and C. On a piece of paper we build some kind of triangle A 1 B 1 C 1, in which angle A 1 \u003d angle A, angle C! \u003d angle C and measure the lengths of the sides A 1 B 1 and A 1 C 1 of this triangle. Since the triangle ABC is similar to the triangle A 1 B 1 C 1, then AB: A 1 B 1 \u003d AC: A 1 C 1, from where we find AB from the known distances AC, A 1 C 1, A 1 B 1. . For the convenience of calculations, it is convenient to construct a triangle A 1 B 1 C 1 so that A 1 C 1: AC \u003d 1: 1000

b) To measure the width of the river on the bank, we measure the distance AC, with the help of an astrolabe we set the angle A = 90 0 (pointing at object B on the opposite bank), measure the angle C. On a piece of paper we build a similar triangle (more conveniently on a scale of 1: 1000) and calculate AB (width of the river).

c) The width of the river can also be determined as follows: considering two similar triangles ABC and AB 1 C 1. Point A was chosen on the bank of the river, B 1 and C at the edge of the water surface, BB 1 - the width of the river (reference No. 583, Fig. 204 of the textbook), while measuring AC, AC 1, AB 1.

Practical work: determine the height of the tree, the width of the river.

In grade 9, in paragraph 100, measurement work on the ground is also considered, but the topic “Solving triangles” is used, while the sine theorem and the cosine theorem are applied. Problems with specific data are considered, solving which you can see various ways of finding and height of an object and determine the distance to an inaccessible point, which can be applied in practice in the future.

1. Measuring the height of an object.

Let's assume that it is required to determine the height AH of some object. To do this, mark point B at a certain distance a from the base H of the object and measure the angle ABH. According to these data, from the right-angled triangle AHB we find the height of the object: AH = HB tgABH.

If the base of the object is not available, then you can do this: on a straight line passing through the base H of the object, mark two points B and C at a certain distance a from each other and measure the angles ABH and DIA: angle ABH = a, angle ASV = b, angle BAC = a-b. These data allow you to determine all the elements of the triangle ABC; by the sine theorem we find AB:

AB \u003d sin ( a-b). From the right triangle ABH we find the height AN of the object:

AH = AB sin a.

№ 1036

The observer is at a distance of 50 m from the tower whose height he wants to determine. He sees the base of the tower at an angle of 10 0 to the horizon, and the top - at an angle of 45 0 to the horizon. What is the height of the tower? (Fig. 298 textbook)

Decision

Consider the triangle ABC - rectangular and isosceles, because the angle CBA = 45 0, then the angle BCA = 45 0, which means CA = 50m.

Consider the triangle ABH - right-angled, tg (ABH) = AH / AB, hence

AN \u003d AB tg (ABN), i.e. AN \u003d 50tg 10 0, hence AN \u003d 9m. CH \u003d SA + AN \u003d 50 + 9 \u003d 59 (m)

№ 1038

There is a tower on the mountain, the height of which is 100m. Some object A at the foot of the mountain is observed first from the top B of the tower at an angle of 60 0 to the horizon, and then from its base C at an angle of 30 0 . Find the height H of the mountain (figure 299 of the textbook).

Decision:

angle EBA = 60 0

KSA angle =30 0

Find SR.

Decision:

SVK angle = 30 0, because angle EBC \u003d 90 0 and angle EBA \u003d 60 0, hence the angle SKA \u003d 60 0, then the angle SKA \u003d 180 0 - 60 0 \u003d 120 0.

In the triangle SKA we see that the angle ASK = 30 0 , the angle SKA = 120 0 , then the angle SAK = 30 0 , we get that the triangle BCA is isosceles with the base AB, because angle SVK = 30 0 and angle BAC = 30 0, which means AC = 100m (BC = AC).

Consider a triangle ASR, a right-angled triangle with an acute angle of 30 0 (RAS = ASC, lying crosswise angles at the intersection of parallel lines SC and AR of the secant AC), and opposite the angle of 30 0 lies the leg half the hypotenuse, therefore PC = 50m.

2. Measuring the distance to an inaccessible point (measuring the width of the river).

Case 1 Measuring the distance between points A and B separated by an obstacle (river).

We choose two accessible points A and B on the river bank, the distance between which can be measured. From point A, both point B and point C, taken on the opposite bank, are visible. Let's measure the distance AB, with the help of an astrolabe we measure the angles A and B, the angle DAB \u003d 180 0 - angle A - angle B

Knowing one side of the triangle and all the angles, by the sine theorem we find the desired distance.

2nd case.

Measuring the distance between points A and B separated by an obstacle (lake). Points A and B are available.

A third point C is selected from which points A and B are visible and the distances to them can be directly measured. It turns out a triangle, in which the angle DAB (measured using an astrolabe) and the sides AC and BC are given. Based on these data, using the cosine theorem, you can determine the size of the side AB - the desired distance. AB 2 \u003d AC 2 + BC 2 - 2 AC * BC cos angle C.

3 case:

Measuring the distance between points A and B, separated by an obstacle (forest) and inaccessible to the determiner of the distance (the points are on the other side of the river).

Two available points C and K are selected, the distance between which can be measured and from which both point A and point B are visible.

The astrolabe is set up at point C and the angles ASC and BSC are measured. Then the distance SK is measured and the astrolabe is transferred to point K, from which the angles AKS and AKB are measured. On paper, along the side SK, taken on a certain scale and two adjacent angles, triangles ASK and VSK are built and the elements of these triangles are calculated. Having drawn the line AB on the drawing, determine its length directly from the drawing or by calculation (they solve the triangles ABC and ABK, which include the line AB being determined).

Practical work in grade 9 at geometry lessons:

  • measure the height of an object;
  • distance to an inaccessible point (river width).

To carry out the work both through the similarity of triangles and through the topic “Solution of triangles”.

Task: compare the results.

As a result of a cycle of lessons on the consideration of the practical application of geometry, students are convinced of the direct application of mathematics in the practical life of a person (measuring the distance to an inaccessible point, determining the height of an object in various ways by the end of education in basic school, using measuring instruments). Solving problems of this type arouses the interest of students who are looking forward to lessons related to direct measurement on the ground. And the tasks proposed in the textbook introduce various ways of solving these problems.

Literature:

  1. Atanasyan L.S. Geometry 7 -9. - Moscow: Enlightenment, 2000

In the early stages of its development, geometry was a set of useful but unrelated rules and formulas for solving problems that people faced in everyday life. Only many centuries later, scientists Ancient Greece was created theoretical background geometry.

AT ancient times The Egyptians, starting to build a pyramid, a palace or an ordinary house, first marked the directions of the sides of the horizon (this is very important, since the illumination in the building depends on the position of its windows and doors in relation to the Sun). They acted like this. They stuck a stick vertically and watched its shadow. When this shadow became the shortest, then its end indicated the exact direction to the north.

egyptian triangle

To measure the area, the ancient Egyptians used a special triangle, which had fixed side lengths. Special specialists, who were called "rope tensioners" (harpedonaptai), were engaged in measurements. They took a long rope, divided it into 12 equal parts with knots, and tied the ends of the rope. In the north-south direction, they set up two stakes at a distance of four parts marked on the rope. Then, with the help of the third stake, the tied rope was pulled so that a triangle was formed, in which one side had three parts, the other four, and the third five parts. A right-angled triangle was obtained, the area of ​​which was taken as a standard.

Determination of inaccessible distances

The history of geometry contains many methods for solving problems of finding distances. One of these tasks is to determine the distances to ships at sea.

The first method is based on one of the signs of equality of triangles

Let the ship be at point K, and the observer at point A. It is required to determine the distance of the spacecraft. Having built a right angle at point A, it is necessary to lay two equal segments on the shore:

AB = BC. At point C, again construct a right angle, and the observer must go along the perpendicular until he reaches point D, from which the ship K and point B would be seen as lying on the same straight line. Right-angled triangles BCD and BAK are equal, therefore, CD = AK, and the segment CD can be directly measured.

The second way is triangulation

It was used to measure distances to celestial bodies. This method includes three steps:

□ Measure angles α, β and distance AB;

□ Construct a triangle А1 В1К1 with angles α and β at the vertices А1 and В1 respectively;

□ Given the similarity of triangles ABK and A1 B1K1 and the equality

AK: AB \u003d A1K1: A1 B1, from the known lengths of the segments AB, A1K1 and, A1 B1, it is easy to find the length of the segment AK.

The technique used in Russian military instructions at the beginning of the 17th century.

Task. Find the distance from point A to point B.

At point A, you need to choose a wand about the size of a person. The upper end of the rod should be aligned with the top of the right angle of the square so that the extension of one of the legs passes through point B. Next, you need to mark the point C of the intersection of the extension of the other leg with the ground. Then, using the proportion

AB: AD = AD: AC, easy to calculate the length of AB; AB = AD2 / AC. In order to simplify calculations and measurements, it is recommended to divide the wand into 100 or 1000 equal parts.

An ancient Chinese technique for measuring the height of an inaccessible object.

A huge contribution to the development of applied geometry was made by the largest Chinese mathematician of the 3rd century, Liu Hui. He owns the treatise "Mathematics of the Sea Island", which provides solutions to various problems for determining the distances to objects located on a remote island, and calculating inaccessible heights. These tasks are quite difficult. But they have practical value, so they got wide application not only in China, but also abroad.

Watching the sea island. To do this, they installed a pair of poles of the same height of 3 zhang at a distance of 1000 bu. The bases of both poles are on the same line with the island. If you move in a straight line from the first pole to 123 bu, then the eye of a person lying on the ground will observe the upper end of the pole coinciding with the top of the island. The same picture will turn out if you move away from the second pole by 127 bu.

What is the height of the island?

In the notation familiar to us, the solution of this problem, based on the properties of similarity.

Let EF = KD = 3 zhang = 5 bu, ED = 1000 bu, EM = 123 bu, CD = 127 bu.

Define AB and AE.

Triangles AVM and EFM, ABC and DKS are similar. Therefore, EF:AB = EM:AM and KD:AB = DC:AC. We get: EM:AM = DC:AC, or EM: (AE + EM) = CD: (AE + ED + DC). As a result, we find AE \u003d 123 1000: (127 - 123) \u003d 30750 (bu). Triangles A1BF and EFM are also similar, and AB = A1B + A1A. Hence AB \u003d 5 1000 (127 - 123) + 5 \u003d 1255 (bu)

How to find the height of an island?

□ Multiply the height of the pole by the distance between the poles - this is the divisible.

□ The difference between the offsets will be the divisor, divide by it.

□ What happens, add the height of the pole.

□ Get the height of the island.

The recipe suggested by Liu Hui.

Distance to an inaccessible point.

❖ The deviation from the previous pole multiplied by the distance between the poles is the divisible.

❖ The difference between the waste will be the divisor, divide by it.

❖ Get the distance the island is from the pole.

Applied geometry was indispensable for land surveying, navigation and construction. Thus, geometry has accompanied mankind throughout the history of its existence. The solution of certain ancient problems of an applied nature can still be applied at the present time, and therefore deserve attention today.

Ministry of Education and Science of the Republic of Khakassia

Municipal educational institution

Ustino-Kopyovskaya secondary school.

Section of mathematics.

MEASUREMENT WORKS ON THE TERRAIN

VILLAGE Ustinkino

Supervisor: Romanova

Elena Alexandrovna,

mathematic teacher

Ustinkino, 2010

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

1. The emergence of measurements in antiquity

1.1 Units of measurement of different peoples…………………………………..4

5

1.3 Geometry in ancient practical problems…………………………..7

1.4 Instruments for measuring on the ground……………………………………………………7

2. Measurement work on the ground

2.1 Construction of a straight line on the ground (suspension

straight line)…………………………………………………………...8

2.2 Measuring average stride length………………………………………..9

2.3 Construction of right angles on the ground…………………………………9

2.4 Constructing and measuring angles using an astrolabe……………...10

2.5 Constructing a circle on the ground………………………………...10

2.6 Measuring the height of trees……………………………………….......11

3. Results of measurements on the ground…………………………………………..

3.1 Site planning

3.2 Trees are a threat to life

3.3 Reference - proposal to the Village Council p. Ustinkino

Conclusion……………………………………………………………………………21

Literature…………………………………………………………………………….22

Introduction

To make a model of figures, I had to perform more than 20 different operations. And almost half of them are related to measurements. I wonder if there are professions in which nothing needs to be measured with instruments at all. I haven't found any. I was not able to find a school subject, in the study of which there would be no need for measurements.


"Science begins when

How to start measuring

Exact science is unthinkable

without measurement.

Indeed, the role of measurements in the life of modern man is very great.

The popular encyclopedic dictionary defines a dimension. Measurements are actions performed in order to find numerical values, quantitative quantities in accepted units of measurement. ¹

You can measure the value with the help of instruments. In everyday life, we can no longer do without a watch, a ruler, a measuring tape, a measuring cup, a thermometer, an electric meter. We can say that we encounter devices at every step.

Purpose: study of geometric measurements on the terrain with. Ustinkino.

study the history of the emergence of measurements;

familiarize yourself with and make devices for measuring on the ground;

make measurements on the ground;

draw conclusions and formulate their proposals.

Hypothesis: at present, measuring work on the ground plays an important role, since without taking measurements you can pay with your life.

Object of study: measurements on the ground.

Subject of study: methods of measurements on the ground.

___________________________________

21 . Popular encyclopedic dictionary. Scientific publishing house "Great Russian Encyclopedia". Publishing house "ONIX 21st century", 2002, p. 485

1. The emergence of measurements in antiquity

In ancient times, a person had to gradually comprehend not only the art of counting, but also measurements. When ancient man, already thinking, tried to find a cave for himself, he was forced to measure the length, width and height of his future home with his own height. And this is what measurement is. Making the simplest tools, building houses, getting food, it becomes necessary to measure distances, and then areas, capacities, mass, time. Our ancestor had only his own height, the length of arms and legs. If a person used fingers and toes when counting, then hands and feet were used to measure distances. There was no people who would not have invented their own units of measurement.

1.1 Units of measurement of different peoples

The builders of the Egyptian pyramids considered the cubit (the distance from the elbow to the end of the middle finger) as the standard of length, the ancient Arabs - the hair from the muzzle of a donkey, the British still use the royal foot (translated from English, "foot" means "leg"), equal to the length king's feet. The foot length has been refined with the introduction of the stock unit. This is "the length of the feet of 16 people leaving the church from Sunday morning." Dividing the length of the stem into 16 equal parts, we got the average length of the foot, because people left the church different height. The length of the foot became 30.48 cm. The English yard is also associated with dimensions human body. This measure of length was introduced by King Edgar and was equal to the distance from the tip of His Majesty's nose to the tip of the middle finger of the outstretched hand. As soon as the king changed, the yard lengthened, as the new monarch was of a larger build. Such changes in length caused great confusion, so King Henry I legalized a permanent yard and ordered a standard to be made from elm. This yard is still used in England (its length is 0.9144 m). To measure small distances, the length of the joint of the thumb was used (translated from the Dutch "inch" means " thumb"). The length of an inch in England was refined and became equal to the length of three grains of barley, taken out of the middle part of the ear and set to each other with their ends. It is known from English novels and stories that peasants often determined the height of horses with their palms.


To measure large distances in antiquity, a measure was introduced, called a field, and then a verst appears instead of it. This name comes from the word "twirl", which at first meant the turn of the plow, and then - a row, the distance from one to another turn of the plow during plowing. The length of a verst at different times was different - from 500 to 750 fathoms. Yes, and there were two versts: a track - it measured the distance of the path and a boundary - for land plots.

The distance was measured in steps by almost all peoples, but for measuring fields and other large distances, the step was too small a measure, so a cane, or double step, was introduced, and then a double cane, or persha. In maritime affairs, a cane was called a stock. In England there was such a measure as a good plowman's stick, the length of which was 12 - 16 feet. In Rome, a measure is introduced equal to a thousand double steps, called a mile (from the word "mille", "milia" - "thousand").

The Slavs had such a measure of length as "throwing a stone" - throwing a stone, "shooting" - the distance that an arrow flew from a bow. Distances were measured like this: “Pechenegia was five days away from the Khazars, six days from the Alans, one day from Russia, four days from the Magyars, and half a day’s journey from the Danube Bulgarians.” In ancient letters of grant of land, you can read: "From the churchyard in all directions to the roar of a bull." This meant - at a distance from which the roar of a bull can still be heard. Other peoples had similar measures - “cow cry”, “cock cry”. Time also served as a measure - "until the boiler of water boils." Estonian sailors said that there were still “three pipes” to the shore (the time spent smoking pipes). "Cannon shot" is also a measure of distance. When horseshoes for horses were not yet known in Japan and they were shod with straw soles, the “straw shoe” measure appeared - the distance at which this shoe wore out. In Spain, the distance measure "cigar" is known - the distance that a person can walk while smoking a cigar. In Siberia, in ancient times, the distance measure "beech" was used - this is the distance at which a person ceases to see separately the horns of a bull.

3.3 Reference - proposal to the Village Council p. Ustinkino

Chairman of the SS Ustinkino

10th grade students

Solenik Alena

Help offer

I measured the height of electric poles, the height of which is always exactly 17 m. By measuring the height of trees, unexpected results were obtained. Tree heights range from 19 m to 56 m.

I think that it is necessary to pay attention to the height of the trees and in the spring to cut the trees to a height of 19 m.

___________________ __________________

CONCLUSION

This essay discusses the most pressing problems associated with geometric constructions on the ground - hanging straight lines, dividing segments and angles, measuring the height of a tree. Given a large number of problems and their solutions are given. The above tasks are of considerable practical interest, consolidate the knowledge gained in geometry and can be used for practical work.

Thus, the purpose of the essay, I think, has been achieved, the tasks set have been completed. I hope for my help - the proposal will be taken into account and executed according to the requirement.

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