It - ancient geometric problem.

Step-by-step instruction

1st way. - With the help of the "golden" or "Egyptian" triangle. The sides of this triangle have an aspect ratio 3:4:5, and the angle is strictly 90 degrees. This quality was widely used by the ancient Egyptians and other pra-cultures.

Fig.1. Construction of the Golden, or Egyptian Triangle

• We make three measurements (or rope compasses - a rope on two nails or pegs) with lengths of 3; four; 5 meters. The ancients often used the method of tying knots with equal distances between them as units of measurement. The unit of length is " knot».
• We drive in a peg at point O, we cling to it the measurement “R3 - 3 knots”.
• We stretch the rope along the known border - towards the proposed point A.
• At the moment of tension on the border line - point A, we drive in a peg.
• Then - again from the point O, we stretch the measure R4 - along the second border. We do not drive the peg in yet.
• After that, we stretch the measure R5 - from A to B.
• At the intersection of the measurements R2 and R3 we drive in a peg. - This is the desired point B - third vertex of the golden triangle, with sides 3;4;5 and with a right angle at point O.

2nd way. With the help of a circle.

The circle can be rope or in the form of a pedometer. Cm:

Our compass pedometer has a step of 1 meter.

Fig.2. Compass pedometer

Construction - also according to Ill.1.

• From the reference point - point O - the corner of the neighbor, we draw a segment of arbitrary length - but more than the radius of the compass = 1m - in each direction from the center (segment AB).
• We put the leg of the compass at point O.
• We draw a circle with a radius (compass step) = 1m. It is enough to draw short arcs - 10-20 centimeters each, at the intersections with the marked segment (through points A and B.). By this action, we found equidistant points from the center- A and B. The distance from the center does not matter here. You can simply mark these points with a tape measure.
• Next, you need to draw arcs with centers at points A and B, but with a slightly (arbitrarily) larger radius than R = 1m. It is possible to reconfigure our compass to a larger radius if it has an adjustable pitch. But for such a small current task, I would not want to “pull” it. Or when there is no regulation. Can be done in half a minute rope compasses.
• We put the first nail (or the leg of a compass with a radius greater than 1 m) alternately at points A and B. And we draw the second nail - in a tense state of the rope, two arcs - so that they intersect with each other. It is possible at two points: C and D, but one is enough - C. And again, short serifs at the intersection at point C are enough.
• We draw a straight line (segment) through points C and D.
• All! The resulting segment, or straight line, is exact direction on North:). Sorry, - at a right angle.
• The figure shows two cases of boundary mismatch over the neighbor's site. Figure 3a shows the case when the neighbor's fence moves away from the desired direction to the detriment of itself. On 3b - he climbed onto your site. In situation 3a, it is possible to construct two “guide” points: both C and D. In situation 3b, only C.
• Place a peg at corner O, and a temporary peg at point C, and stretch a cord from C to the back of the lot. - So that the cord barely touches the peg O. By measuring from point O - in the direction D, the length of the side according to the general plan, get a reliable rear right corner of the site.

Fig.3. Building a right angle - from the corner of a neighbor, using a pedometer compass and a rope compass

If you have a compass pedometer, then you can do without a rope. Rope in the previous example, we used to draw arcs of a larger radius than the pedometer. More because these arcs must intersect somewhere. In order for the arcs to be drawn with a pedometer with the same radius - 1m with a guarantee of their intersection, it is necessary that points A and B are inside the circle c R = 1m.

• Then measure these equidistant points roulette- in different directions from the center, but always along the AB line (neighbor's fence line). The closer points A and B are to the center, the further away from it are the guide points: C and D, and the more accurate the measurements. In the figure, this distance is taken to be about a quarter of the radius of the pedometer = 260mm.

Fig.4. Constructing a right angle with a pedometer compass and a tape measure

• This scheme of actions is no less relevant when constructing any rectangle, in particular, the contour of a rectangular foundation. You will get it perfect. Its diagonals, of course, need to be checked, but don't efforts decrease? - Compared to when the diagonals, corners and sides of the foundation contour move back and forth until the corners meet ..

Actually, we have solved the geometric problem on the ground. In order for your actions to be more confident on the site, practice on paper - using a regular compass. Which is basically no different.

Often, a home master urgently needs to make some kind of measurement or markup at a certain angle, and there is no square or protractor at hand. In this case, a few simple rules will help him out.

90 degree angle.

If you urgently need to build a right angle, but there is no square, you can use any printed publication. The corner of a paper sheet is a very precise right angle (90 degrees). Cutting (punching) machines in printing houses are set up very precisely. Otherwise, the original roll of paper will start to cut at random. Therefore, you can be sure that this angle is exactly right.

And if there is not even a printed publication or it is necessary to build a corner on the ground, for example, when marking a foundation or a sheet of plywood with jagged edges? In this case, the rule of the golden (or Egyptian) triangle will help us.

A golden (or Egyptian, or Pythagorean) triangle is a triangle with sides that are related to each other as 5:4:3. According to the Pythagorean theorem, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs. Those. 5x5 = 4x4 + 3x3. 25=16+9 and this is indisputable.

Therefore, to build a right angle, it is enough to draw a straight line with a length of 5 (10,15,20, etc., a multiple of 5 cm) on the workpiece. And then, from the edges of this line, start measuring 4 on one side (8,12,16, etc., a multiple of 4 cm), and on the other, 3 (6,9,12,15, etc., a multiple of 3 cm) distances. You should get arcs with a radius of 4 and 3 cm. Where these arcs intersect each other and there will be a straight (90 degrees) angle.

45 degree angle.

Such angles are usually used in the manufacture of rectangular frames. The material from which the frame (baguette) is made is sawn at an angle of 45 degrees and joined. If there is no miter box or protractor at hand, you can get a 45 degree angle template as follows. It is necessary to take a sheet of writing paper or any printed publication and bend it so that the fold line passes exactly through the corner, and the edges of the folded sheet coincide. The resulting angle will be equal to 45 degrees.

Angle of 30 and 60 degrees.

An angle of 60 degrees is required to construct equilateral triangles. For example, you need to file such triangles for decorative work or precisely set the power cut. An angle of 30 degrees is rarely used in its pure form. However, with its help (and with the help of an angle of 90 degrees), an angle of 120 degrees is built. And this is the angle needed to build equilateral hexagons, a figure very popular with carpenters.

To build a very accurate pattern of these angles at any time, you need to remember the constant (number) 173. They follow from the ratio of the sines and cosines of these angles.

Take a sheet of paper from any printed publication. Its angle is exactly 90 degrees. Measure 100 mm (10 cm) from the corner on one side and 173 mm (17.3 cm) on the other side. Connect these dots. Thus, we got a template that has one angle of 90 degrees, one 30 degrees and one 60 degrees. You can check on the protractor - everything is for sure!

Remember this number - 173, and you can always build angles of 30 and 60 degrees.

The squareness of the workpiece.

When marking workpieces or constructions on parts, in addition to the corners themselves, their ratio is also very important. This is especially important in the manufacture of rectangular parts or, for example, when marking the foundation, cutting large sheets of material. Incorrect construction or layout leads subsequently to a lot of unnecessary work or to the appearance of a large amount of waste.

Unfortunately, even very accurate marking tools, even professional ones, always have a certain error.

Meanwhile, there is a very simple method for determining the squareness of a part or construction. In a rectangle, the diagonals are absolutely equal! This means that after construction it is necessary to measure the lengths of the diagonals of the rectangle. If they are equal, it's ok, it's really a rectangle. And if not, you have built a parallelogram or a rhombus. In this case, you should “play” a little with adjacent sides in order to achieve exact (for this case) equality of the diagonals of the rectangle to be marked.

Look at the picture. (Fig. 1)

Rice. 1. Illustration for example

What geometric shapes are familiar to you?

Of course, you saw that the picture consists of triangles and rectangles. What word is hidden in the name of both these figures? This word is an angle (Fig. 2).

Rice. 2. Determining the angle

Today we will learn how to draw a right angle.

The name of this angle already has the word "straight". To correctly depict a right angle, we need a square. (Fig. 3)

Rice. 3. Square

The square itself already has a right angle. (Fig. 4)

Rice. 4. Right angle

He will help us to depict this geometric figure.

To correctly depict the figure, we must attach the square to the plane (1), circle its sides (2), name the vertex of the angle (3) and the rays (4).

1.

2.

3.

4.

Let's determine if there are straight lines among the available angles (Fig. 5). A square will help us with this.

Rice. 5. Illustration for example

Let's find the right angle of the square and apply it to the existing angles (Fig. 6).

Rice. 6. Illustration for example

We see that the right angle coincided with the PTO angle. This means that the PTO angle is right. Let's do the same operation again. (Fig. 7)

Rice. 7. Illustration for example

We see that the right angle of our square did not coincide with the COD angle. This means that the angle COD is not a right angle. Once again we apply the right angle of the square to the angle AOT. (Fig. 8)

Rice. 8. Illustration for example

We see that the AOT angle is much larger than the right angle. This means that the AOT angle is not a right angle.

In this lesson, we learned how to build a right angle using a square.

The word "angle" gave the name to many things, as well as geometric shapes: a rectangle, a triangle, a square, with which you can draw a right angle.

A triangle is a geometric figure that consists of three sides and three angles. A triangle that has a right angle is called a right triangle.

General rules for any foundation

We choose a starting point. The first side of our foundation needs to be tied to some object of our site.

Example. Let's make our foundation (house) parallel to one of the sides of the fence. Therefore, we stretch the first twine equidistant from this side of the fence to the distance we need.

Construction of a right angle (90⁰). As an example, we will consider a rectangular foundation in which all angles are as close as possible to 90⁰.

There are several ways to do this. We will look at 2 main ones. © www.site

Method 1. Golden Triangle Rule

To construct a right angle, we will use the Pythagorean theorem.

In order not to delve into the geometry, let's try to describe it in a simpler way. So that between two segments a and b to make an angle of 90⁰, you need to add the lengths of these segments and take the root of this sum. The resulting number will be our long diagonal connecting our segments. It is very easy to calculate with a calculator.

Usually, when marking the foundation, they take the dimensions of the sides, so that when deduced from the root, an integer is obtained. Example: 3x4x5; 6x8x10.

If you have a tape measure, then in general there will be no problems if you take segments other than those commonly used. For example: 3x3x4.24; 2x2x2.83; 4x6x7.21

If we made measurements in meters, then the values ​​are very clear: 4m24cm; 2m83cm; 7m21cm.

Calculator

2 + 2 = 2

It is also worth noting that measurements can be made in any length measurement systems, the main thing is to use the aspect ratio known to us: 3x4x5 meters, 3x4x5 centimeters, etc. That is, even if you do not have a tool for measuring the length, you can take, for example, a rail (the length of the rail does not matter) and measure it (3 rails x 4 rails x 5 rails).

Now let's see how to put it into practice.

Instructions for marking a rectangular foundation

Method 1. The rules of the golden triangle (t. Pythagoras)

Consider, for example, the construction of a rectangular foundation with dimensions of 6x8m using a golden triangle (t. Pythagoras).

1. We mark the first side of the foundation. This is the easiest part in building our rectangle. The main thing to remember. If we want our foundation (house) to be parallel to one of the sides of the fence or another object on the site or beyond, then we make the first line of our foundation equidistant from the object we have chosen. We have described this procedure above. Pegs firmly planted in the ground can be used to place the first twine, but ideally a scrap piece is used for this purpose. We will use it. The distance between the cast-offs for this side will be 14m: between the cast-offs and the future corners of 3m and 8m under the foundation.

2. We stretch the second string as perpendicular as possible to the first. Ideally perpendicular in practice, it is difficult to pull, so in the figure we also displayed it slightly deviated.

3. We fasten both strings at the intersection point. You can fasten with a bracket or tape. The main thing is to be safe.

4. We proceed to the formation of a right angle using the Pythagorean theorem. We will build a right triangle with legs 3 by 4 meters and a hypotenuse of 5 meters. To begin with, we measure 4 meters on the first string from the intersection of the strings, and on the second 3 meters. We put marks on the lace using adhesive tape (clothespin, etc.).

5. We connect both marks with a tape measure. We fix one end of the tape measure at the 4-meter mark and lead towards the 3-meter mark on the other twine.

6. If we have a right triangle, then both marks should converge at a distance of 5 meters. In our case, the marks did not match. Therefore, we move the twine in our case to the right until the moment when the mark of 3 m coincides with the division of the tape measure by 5 m.

7. As a result, we got a right triangle with a 90⁰ angle between two strings.

8. We do not need more marks and they can be removed.

9. Let's start building a rectangle. We measure on both strings the lengths of the sides of our foundation 6 and 8 meters, respectively. We put marks on the twine.

10. We stretch the third string as perpendicular as possible to the first string. We fasten both strings at a mark of 8 m.

11. We stretch the fourth string as perpendicular as possible to the second string. We fasten both strings to a mark of 6 meters.

12. We make marks on the third twine 6 meters and on the fourth 8 meters.

13. To get a quadrilateral with right angles in our case, it is necessary that both marks on the third and fourth twine coincide. To do this, move both strings until the marks are connected.

14. As a result, if everything is measured correctly, then we should get the correct rectangle. Let's check if it turned out by measuring the diagonals.

15. Measure the lengths of the diagonals. If they are the same, as in our case, we have the correct rectangle. The diagonals are the same length in an isosceles trapezoid. But we know one angle of 90⁰, and there are no such angles in an isosceles trapezoid.

16. Finished layout of a rectangular foundation using the Pythagorean theorem. © www.site

Method 2. Web

A very simple way to make a markup in the form of a rectangle with 90⁰ corners. The most important thing we need is a string that does not stretch, and the accuracy of your measurements with a tape measure.

1. Cut the pieces of twine that we need to form the markup. In this example, we are building a foundation with sides of 6 by 8 meters. Also, for the correct construction of a rectangle, we need equal diagonals, which for a rectangle of 6 by 8 meters will be equal to 10 meters (so Pythagoras is described above). You also need to take a margin of length of twine for fastening.

2. We connect our "web" as in the figure. We fasten the sides with diagonals in 4 places at the corners. The diagonals themselves at the intersection point do not need to be fastened.

3. We stretch the first twine (points 1,2). We will fasten it with pegs. The main thing is that the pegs are firmly held in the ground and that when our structure is pulled, they are not taken away. This important point must be taken into account.

4. We stretch the corner 3. The main condition is that the twine 1-3 and the diagonal 2-3 do not sag and are stretched as much as possible. After fixing with the help of a peg at point 3, we have an angle at point 1 of 90⁰.

5. Stretch corner 4 and set the peg. We make sure that the twine at points 2-4, 3-4 and the diagonal 1-4 do not sag and are as tight as possible.

6. If all the conditions are met, then as a result we should get a rectangle with corners as close as possible to 90⁰.

Marking under the foundation of the house

We make a two-tier cast-off. The lower tier is the level of the pillars.

The upper tier of the cast-off is the level of the grillage.

We create a rectangle for the outer contour using the so-called Pythagoras. Then we retreat by an amount equal to the width of the tape and make an inner contour.

The easiest way to markup. We build a rectangle according to the dimensions of the foundation using the Pythagorean theorem to find the right angle. © www.site

From the author

In this article, we looked at how to mark up the foundation with our own hands with the construction of a rectangle with 90⁰ corners. In general, there is nothing difficult in the markup. The price of the question is the cost of twine, a cast-off board (economy option - pegs) and the ability to use a tape measure.

People who build a country house for the first time on their own often get lost when marking the site. Indeed, it is much more difficult to lay down an angle on the ground or draw a straight line than on paper - the scale is different. The matter is complicated by the fact that a natural site is never perfectly flat and there are always features of the landscape that interfere with measurement. However, the problem is solvable.

The markup is based on the principles of geometry, which initially served this very purpose: the word itself, translated from Greek, means “measurement of the earth”. So laying off corners on the ground is not a new thing, similar to drawing in a school notebook. Nevertheless, the difference is significant: a ruler and a compass are used to build a figure on paper, but you cannot use them on a real site.

How to build a right angle on the ground

A long reinforced thread or a suitable twine (”clothes” rope) will help out in this situation.

With the help of a thread, straight lines and segments are built. To do this, at the starting point, a peg is driven into the ground, to which one end of the thread is tied. Then the thread is pulled in the desired direction, in the case of constructing a segment - to a given length, previously marked on the thread. At the point obtained, a second peg is driven in and, pulling it tightly, a thread is tied to it. If the twine is used only for measurement, then it makes sense to pre-apply a meter scale on it. To do this, every second meter is covered with black paint, preferably waterproof, and every fifth with bright (for example, red). This “zebra” simplifies marking, allowing you to quickly measure long segments. Sometimes it makes sense to make the scale smaller by coloring every 50 or even 20 cm of twine.

If the terrain is very uneven, then it is better to use “suspended” markings, driving in pegs of different heights (Fig. 1, a). If the difference in height between the start and end points is too large (the site is located on a steep slope), then the task becomes a little more complicated. You can use several pegs, summing up the distance between them. True, when marking with “steps”, you need to ensure that the angle between the peg and the rope remains straight. (Fig. 1, b).

In order to lay a right angle on the ground, you can use the principle of a triangle, where the sides are related as 3:4:5 (the so-called “Pythagorean triple”). In this case, the triangle is right-angled, with angles of 90, 60, and 30 degrees. The smaller sides are legs, the angle between them is straight.

In practice, the method is applied as follows. On the ground, from the starting point "0" (see fig. 2), marked with a peg, a straight line is drawn, on which a segment 4 meters long is laid - the side of the future corner (“a”). The end of the segment (point "1") is marked with a peg. Then, a thread is tied to the initial peg, with a mark at a distance of exactly 3 meters from the peg, and it fits on the ground by eye, approximately in the direction of the second side of the corner (”b”). From point 1 to the end of thread b, a thread is laid similarly with a mark at 5 meters (“c”). Then the threads b and c must be taken in different hands, stretched as much as possible and in this state brought together, exactly aligning the marks (point “2″”). The result is a triangle, where the “zero” angle will be right. For clarity, a schematic drawing is given.

The lengths of the guide threads can be larger or smaller, but must necessarily be related as 4:3:5. Obviously, the right angle will always lie opposite the larger side of the triangle.

In the same way, you can easily set aside almost any angle that is a multiple of 30 degrees, choosing the length of the guide threads. Here is the length ratio for some angles: 90 degrees (a = 4; b = 3; c = 5), 60 degrees (a = 3; b = 5; c = 4 or a = 5; b = 5; c = 6) , 30 degrees (a = 5; b = 4; c = 3), 120 degrees (a = 5; b = 5; c = 8)

How to correctly calculate a right angle

How to find a 90 degree right angle

How to find a 90 degree angle using a construction tape measure and a pencil?

Many builders faced such a problem - how to find an angle of 90 degrees or how to find out if an angle is obtuse (greater than 90 degrees) or acute (less than 90 degrees).

We will not return to school geometry and study tricky words, but consider in practice, where each person, literally in one minute, can determine how many degrees this or that angle has. And in 5 minutes, you can make an exact square with a right angle, that is, 90 °.

Let's take for example.
On one side (on the leg “a”) we measure 60 cm. Then on the other side (leg “b”) we measure 80 cm. If from the point “a” to the point “b” the perpendicular “c” will be 100 cm (1 meter ) so the angle is 90 degrees. If more, for example 1.1 m, the angle is obtuse, and when 0.9 m, the angle is acute. Thus, with the help of a construction tape measure and a pencil, we were able to get a right angle.

Now let's analyze the numbers 60 and 80 and why the perpendicular should have 1 m. We take a combination of numbers “3,4,5” and multiply each number by our invented number - for example, “5”.

3 (multiply) 5 \u003d 15 legs
4*5=20 legs
5*5=25 hypotenuse

In the above example, we took the numbers “30, 40, 50” and multiplied each number by “2”, in this way, we got the following combination:
30*2=60 legs
40*2=80 legs
50*2=100 hypotenuse

How to make a 45 degree angle with a construction tape measure and a pencil?

Before you get a 45 degree angle, use the above system to make a right angle. Then, on the leg “a” and “b”, we measure the same sizes and draw the hypotenuse. Measure the hypotenuse and divide by two (/2). Then we draw a line to the right angle. In this way, we divided 90 degrees into 45 - two identical parts of 45 degrees.

How to make your own square with a right angle in 5 minutes?

1 We connect two even wooden slats together, so that one of them is perpendicular to the other.

2 Then we measure two legs according to the above system.

3 Arriving wooden rail to the first mark

4 We measure the hypotenuse and fix it on the second leg.

5 We check all dimensions and additionally fix them in all places.

6 Then cut off the excess parts.

How to find a right angle 90 degrees video

How to make a right angle between walls.

Ancient Greek geometers and, in particular, Euclid, tried in vain, their knowledge never reached the Soviet builders. In the sense that there are no rectangular rooms in Soviet houses. And they are at best in the form of a parallelogram, a truncated trapezoid or a rhombus, and at worst and most common in the form of an irregular quadrilateral. This quite often complicates the quality finishing of the premises. You have to find the right angle yourself. In general, this is easy to do.

Marking is easiest to do on the floor. For this you will need:

• Marker, chalk or pencil
• Building level, harsh thread or construction cord.
• Roulette.

Using a building level or a plumb line (easier - using a level, more precisely - using a plumb line), determine the protruding sections of the walls. In these places, transfer the vertical marks to the floor. Draw straight lines through 2 marks along each wall so that the rest of the marks (if you have them) remain between the line and the wall.

If the walls are perpendicular, this distance should be equal to

1.414 m is more accurate than 1.41421356 m, but you don't need that accuracy.

If the distance (hypotenuse of the triangle) is greater, then instead of a right angle between the walls, you have an obtuse one. In order to get a right angle, attach the beginning of the tape measure to the intersection point of the lines in the corner and draw a small arc with a radius of 1 m. Then attach the beginning of the tape measure to the mark on the line along the wall taken as the basis and draw a small arc with a radius of 1.414 m arcs and the intersection point of the lines in the corner of a straight line. This new line will be the outline of the wall. If this is too difficult for you, then simply measure 1.414 m on the hypotenuse from the mark at the wall that you took as a basis. Draw a straight line through the resulting mark and the intersection point of the lines in the corner. In this case, you will get not a right angle, but still much closer to a straight line than the one that was.

How to calculate right angle

If the lines forming the angle are drawn on paper, then you can determine that the angle is right, for example, using a protractor. Attach it parallel to either side so that the zero mark coincides with the top of the corner. If the other side of the angle corresponds to the ninety-degree division of the protractor, then you can be congratulated - you have determined that this particular angle is right. The same can be done with a square, and if absolute accuracy is not required, then even using other items at hand - a matchbox, a floppy disk, a plastic CD / DVD box, and any other rectangular object.

If in the conditions of the problem the lengths of the sides of the triangle are given, then you should determine the one that is the hypotenuse - the angle lying opposite it will be right. The hypotenuse is always the longest side of a right triangle, so there will be no problem with pre-determining it.

Marking the foundation for the house. Forum members say

If there are two such, then the triangle is not rectangular and the angle you need is not in it at all. Otherwise, make an additional check - the square of the length of the hypotenuse must be equal to the sum of the squares of the lengths of the two short sides (legs). If so, then the angle opposite the long side (usually denoted by the letter γ) is right.

If you need to calculate the construction of a right angle, then do the reverse operation described in the previous step. First determine the lengths of the two sides that will form this angle. It is easier to work with a regular isosceles triangle, so it is better to take the same length of the legs. If the result needs to be displayed on paper, then set aside the desired length on the compass, put a point at the top of the future angle and mark it with the letter A. Draw a circle centered at this point and draw a radius, marking the point of contact with the circle with the letter B. Then calculate the length of the hypotenuse - multiply the length of the leg by the square root of two. Put the resulting value on the compass and draw a second circle centered at point B. Then connect the intersection point of the two circles (point C) with the center of the first circle (point A). This will be the right angle YOU.

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Video lesson "Construction of right angles on the ground" - video material that can be used by a teacher in a geometry lesson to familiarize himself with the methods of constructing angles on the ground. This material contains information about the design of the measuring tool - eker, as well as a detailed description of how this device measures angles on the ground. The material reveals the practical application of the subject, connects geometry with the spheres of human life.

We carry out the exact marking of the foundation ourselves

This information causes a great enthusiasm for the subject of study, helps to better assimilate the educational material.

The use of video tools makes it possible to get acquainted with the device device without resorting to additional equipment to demonstrate the device, its device and the principle of operation. When studying the topic of the same name, the video material can become an assistant to the teacher, replacing his story about the device and the operation of the device with a clear detailed description with a voice explanation. Also, this material can be recommended for self-study with in-depth study of the material, as well as simply supplementing a geometry lesson or extracurricular activities in mathematics with cognitive information.

The video lesson begins with the announcement of the title of the topic "Construction of right angles on the ground." The student is informed that special devices are used to build angles on the ground. Among such devices, the simplest measuring device eker is considered. The screen displays a drawn eker, which consists of two bars, the angle between which is 90°. This device is mounted on a tripod, for it to take a stable position. The device is supplemented with nails driven into its bars so that the angle between the straight lines drawn through them will be right, that is, these lines are perpendicular to each other.

The construction of straight lines, the angle ∠АOB between which is 90°, begins with the correct position of the device. Eker is installed in such a way that the plumb line located in its center is located directly above the point that is the top of the corner. The direction of one of the bars follows the direction of one side of the corner. You can fix this direction by installing a milestone that fixes the passage of the OA side. To build a right angle, a milestone is also affixed in the direction of the second bar, fixing the direction of the straight line. Thus, a right angle is obtained, the construction of which is determined by the established milestones.

This device is imperfect, it is the simplest tool for constructing angles on the ground, so students are shown a special device, the use of which is widespread in construction and architecture - this is a theodolite.

The video lesson "Construction of right angles on the ground" is recommended as a visual aid for conducting a lesson on the topic of the same name. It can also be used as an addition to extracurricular work in mathematics, for distance learning, for self-study of the material.

Usually a straight line along one of the 2 widest walls is taken as the basis if there are no other reference points. In this case, the area of ​​\u200b\u200bthe room during further finishing will be reduced to a minimum.

Measure from one of the corners with a tape measure 1 m and put a mark on the line. Do the same on a perpendicular (maybe not quite) line.

Connect the resulting marks so that you get a triangle.

Measure the distance between the received marks.

If the walls are perpendicular this distance should be ~ 1.414 m, more precisely 1.41421356 m, but you won't need this accuracy.

If the distance (hypotenuse of the triangle) is greater, then instead of a right angle between the walls, you have an obtuse one.

How to build a right angle?

In order to get a right angle, attach the beginning of the tape measure to the intersection point of the lines in the corner and draw a small arc with a radius of 1 m. Then attach the beginning of the tape measure to the mark on the line along the wall taken as the basis and draw a small arc with a radius of 1.414 m arcs and the intersection point of the lines in the corner of a straight line. This new line will be the outline of the wall. If this is too difficult for you, then simply measure 1.414 m on the hypotenuse from the mark at the wall that you took as a basis. Draw a straight line through the resulting mark and the intersection point of the lines in the corner. In this case, you will get not a right angle, but still much closer to a straight line than the one that was.

If the distance (hypotenuse of the triangle) is less, then instead of a right angle between the walls, you have a sharp one. In order to get a right angle, step back from the mark on the line along the wall, taken as the basis, a few centimeters. Draw small arcs on the floor according to the principle outlined in the previous paragraph. The resulting line can be moved closer to the wall. The main condition is that the marks of the protruding sections of the wall must remain between the new line and the wall.

If you do not quite understand this text, then the picture will help you better understand:

From the received 2 sides of the rectangle, the remaining 2 sides are determined by the method of parallel transfer.

What angle do the walls form? The first way is measurement.

To design furniture, we not only need to measure the length and height of the walls in an apartment or house, but also need to measure the angle at which the furniture will be installed.

Why should this be done? - so that there are no problems with installation, in order to avoid huge side gaps, and so that the necessary adjustments can still be made in production.

For example, a deployed corner will not allow you to mount a corner kitchen without additional undercuts of the internal corner modules and countertops. An acute corner can pull the furniture body out of the dimensions of the installation dimensions, because it is impossible to install a furniture module into the right corner.

Actually, when the reasons are clarified and the need to measure the angle is obvious - it's up to the small - to measure the angle.
If you have a goniometer in your home arsenal, then no problem, and if not, then the method described below will always come to the rescue.

The first thing to do is to mark two points on the walls at the same level (at the height where the furniture module will be installed) as follows:

• From the corner with a tape measure, measure the size, for example, 500mm, along the left and right wall. and put points.
• Next, measure the diagonal - i.e. distance between points.

So, for example, we have three sizes - leg 500mm., 500mm. and a diagonal of 700mm.

The next step is to build a corner on a template from any material. In our case, I will show how to do this in the autocad program, but you can also do it with a compass, ruler, protractor and material for the template.

1. We draw a horizontal line of 500 mm. with AB dots. (See drawing below.)
2. Draw a circle with a radius of 500mm. centered on point "B".
3. We draw a second circle with a radius of 700mm. centered on point "A".
4. At the point of intersection of the circles, put the point "C".
5. We connect the points "B" and "C" with a segment and get our angle.
6. Then it remains to measure the angle with a protractor on a template or with a special tool in the autocad program. and apply the existing drawing for design.

When the drawing is built, we can finally conclude that the measured angle is 89 degrees, the angle is sharp and it will not be able to negatively affect the installation of furniture, because.

How to accurately mark a right angle on the ground without a protractor?

1 degree is pretty small.

What angle do the walls form? The second way is calculation.

1. We measure 1000 mm from the corner (the more, the better - the error is less ... of course, if you are for a shelf of 400 * 400 mm, then you don’t need to measure more than 400 mm) on both walls, and put marks (if the wallpaper can be with needles);
2. We measure the distance between the marks (it is better to do this together, again for reasons of accuracy), let's say we got 1500 mm.

Those. according to the example it is: (10002+ 10002– 15002) / (2 1000 1000) = -0.125 hence arccos (-0.125)= 97.18 degrees.

Auxiliary information.

The user Nastya Galkina asked a question in the Other Education category and received 11 replies.

How to build a right angle?

There is a method for constructing a right angle using a compass and ruler. First you need to draw a circle with a compass and draw its diameter. Then mark an arbitrary point on the circle and connect it to the ends of the diameter: you get a triangle inscribed in the circle. Its corner (with its apex at a point on the circle) will be a right angle. The second way is to draw any two intersecting circles. Connect the two intersection points with one line, draw the other through the centers of the circles. These two segments will intersect at an angle of 90 degrees. If there are no drawing tools, you can use any rectangular objects. It can be a sheet of cardboard, any packaging (for medicine, a pack of cigarettes, a box of chocolates, etc.), a book, a photo frame, etc.

How to draw a right angle using a compass and straightedge

How to build a right angle?

Before you learn how to build a right angle, you need to remember its definition. A right angle is a ninety degree angle formed by two perpendicular lines. You can also say that this is half of the unfolded angle. There are several ways to construct a right angle.

Ways to construct a right angle

The simplest is the construction of a right angle using a drawing square. It is applied to paper and lines are drawn along perpendicular sides: a right angle is obtained. You can also use a protractor. Attach a protractor to the line drawn with a pencil, mark an angle of ninety degrees on paper. Then connect the line (along the ruler) this mark with the line on paper.

There is a method for constructing a right angle using a compass and ruler. First you need to draw a circle with a compass and draw its diameter. Then mark an arbitrary point on the circle and connect it to the ends of the diameter: you get a triangle inscribed in the circle.

How to mark the foundation. Do-it-yourself construction life hack

Its corner (with its apex at a point on the circle) will be a right angle. The second way is to draw any two intersecting circles. Connect the two intersection points with one line, draw the other through the centers of the circles. These two segments will intersect at an angle of 90 degrees. If there are no drawing tools, you can use any rectangular objects. It can be a sheet of cardboard, any packaging (for medicine, a pack of cigarettes, a box of chocolates, etc.), a book, a photo frame, etc.

Construction of right angles on the ground

In general, the construction of right angles on the ground is necessary in construction, when dividing plots of land, etc. For this, special devices are used - eker, astrolabe, theodolite. But, it is unlikely that these tools will be, for example, at their summer cottage. Then you can use the method used since ancient times. You will need three pegs and ropes of 3, 4 and 5 meters. Stick a peg into the ground, tie ropes of 3 and 4 meters to it, and the rest of the stakes to their ends. Connect the last two pegs with a 5-meter rope, pull the resulting triangle, and hammer these stakes into the ground. The angle of the triangle with the first peg will be right.

As you can see, there are many simple ways to construct a right angle.

How to draw a right angle using a compass and straightedge

How to construct an angle using a compass and a ruler, knowing the tangent of this angle?

First, let's remember what a tangent is.

With the help of a compass and a regular ruler (without divisions), we construct two perpendicular lines

Construct an angle whose tangent is 2/3.

Let us measure an arbitrary segment with a compass and from the point of intersection we set aside up two times, then to the left three times. Let's draw a ray through these points, as shown in the figure. The corner is built.

We construct an angle whose tangent is equal to the cube root of three.

Find this number with a calculator

Let's round up to a convenient value of 1.25 and write it as an improper fraction 5/4. Similar to the previous method with With the help of a compass set aside five identical segments up and four to the left. FROM With the help of a ruler let's pass a beam through them. The corner is built.

Let's construct an angle whose tangent is equal to Π .

And everything is the same as in the previous examples - 19 segments up and six to the left, connected - and the corner is built.

I want to add - due to the fact that I changed the values ​​​​a little, the result of constructing the corners was Little error, but with the naked eye and even with the help of a protractor, it will be invisible.

You can easily check - we take a calculator

And about the correctness of constructing the angle according to the method that I indicated - using a computer program, we build angles according to the given parameters, then we build according to my method - we compare and make sure who is right and who is wrong. - more than a month ago

As you know, by the ratio of the sides of a right triangle, you can find all these trigonometric quantities. In particular, the tangent of an angle is defined as the ratio of the length of the leg (side) lying opposite the given angle and the side adjacent to the given angle. Therefore, the procedure will be as follows:

1) draw any straight line;

2) we draw another line at a right angle to it - for this we draw a circle of any radius with a center located on the first straight line, and then another circle of the same radius with a center located at the intersection point of the first circle and the first straight line; a straight line drawn through two points of intersection of these circles will be perpendicular to the first;

3) from the point of intersection of the first and second straight lines - the vertex of the right angle - we measure a segment of any suitable length on the first straight line, we consider that this is an adjacent leg;

4) knowing the ratio - tangent, we calculate the length of the second leg segment - opposite, (we multiply the tangent by the length of the first segment), and measure it from the same point / vertex on the second straight line;

5) we connect all the vertices of the resulting right triangle, one of the corners of which, with a side on the first line, is the desired one.

FEBUS, I understand, it seems that you mean - with tgA \u003d π, the angle turns out to be close to 90 degrees, and if the tangent of the angle tends to infinity - in general, the length of the ruler to build such a triangle should also be infinite. So what, exactly? The length of one leg will be 3.14 times greater than the length of the other - such a triangle can be constructed using the indicated method. What's wrong? - more than a month ago

The tangent is the ratio of the leg opposite the corner to the leg adjacent to the corner.

The tangent must be represented as a fraction of the numerator (this is the value of the opposite leg) and the denominator (the value of the adjacent leg)

We draw a straight line and draw a perpendicular to it, the intersection point is the vertex of the right angle (point A)

From the point of intersection (the vertex of the right angle - point A) on the straight line, a segment equal to the value of the opposite leg (point B) must be set aside.

On a straight line, it is necessary to postpone a segment equal to the size of the adjacent leg (point C)

We connect points B and C, we get a triangle ABC

The tangent of the angle DIA is equal to the known tangent.

Express as a fraction tgA = π. - more than a month ago

To build an angle with a given value of the tangent of the angle, a compass is not needed, one ruler is enough.

In the coordinate system, we set aside the unit along the abscissa (X), and the value of the tangent of the angle along the ordinate (Y). We connect a point with such coordinates to the origin of the coordinate system. The angle between the X-axis and the constructed line is the desired angle.

Tangent \u003d ratio of the opposite leg to the adjacent one, i.e. tg (a) \u003d Y / X.

I have X = 1, so tg (a) = Y. - more than a month ago