The ability to divide any angle with a bisector is needed not only to get an “A” in mathematics. This knowledge will be very useful for builders, designers, surveyors and dressmakers. In life, you need to be able to divide many things in half. Everyone at school...
Conjugation is a smooth transition from one line to another. To find a mate, you need to determine its points and center, and then draw the corresponding intersection. To solve such a problem, you need to arm yourself with a ruler...
Conjugation is a smooth transition from one line to another. Conjugates are very often used in a variety of drawings when connecting angles, circles and arcs, and straight lines. Construction of a section - quite not an easy task, for which you…
When constructing various geometric shapes, it is sometimes necessary to determine their characteristics: length, width, height, and so on. If we're talking about about a circle or circle, you often have to determine its diameter. The diameter is...
A triangle is called a right triangle if the angle at one of its vertices is 90°. The side opposite this angle is called the hypotenuse, and the sides opposite the two acute angles of the triangle are called the legs. If the length of the hypotenuse is known...
Tasks to construct regular geometric shapes train spatial perception and logic. Exists a large number of very simple tasks of this kind. Their solution comes down to modifying or combining already...
The bisector of an angle is a ray that begins at the vertex of the angle and divides it into two equal parts. Those. To draw a bisector, you need to find the midpoint of the angle. The easiest way to do this is with a compass. In this case you don't need...
When building or developing home design projects, it is often necessary to build an angle equal to an existing one. Templates come to the rescue school knowledge geometry. Instructions 1An angle is formed by two straight lines emanating from one point. This point...
The median of a triangle is a segment connecting any of the vertices of the triangle with the midpoint of the opposite side. Therefore, the problem of constructing a median using a compass and ruler is reduced to the problem of finding the midpoint of a segment. You will need-…
A median is a segment drawn from a certain corner of a polygon to one of its sides in such a way that the point of intersection of the median and the side is the midpoint of that side. You will need - a compass - a ruler - a pencil Instructions 1 Let the given...
This article will tell you how to use a compass to draw a perpendicular to this segment through a certain point lying on this segment. Steps 1Look at the segment (straight line) given to you and the point (denoted as A) lying on it.2Install the needle...
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This article will tell you how to construct a bisector given angle(bisector is a ray dividing an angle in half). Steps 1Look at the angle given to you.2Find the vertex of the angle.3Place the compass needle at the vertex of the angle and draw an arc intersecting the sides of the angle...
This - oldest geometric problem.
Step-by-step instruction
1st method. - Using the “golden” or “Egyptian” triangle. The sides of this triangle have the aspect ratio 3:4:5, and the angle is exactly 90 degrees. This quality was widely used by the ancient Egyptians and other ancient cultures.
Ill.1. Construction of the Golden or Egyptian Triangle
- We manufacture three measurements (or rope compasses - a rope on two nails or pegs) with lengths 3; 4; 5 meters. The ancients often used the method of tying knots with equal distances between them as units of measurement. Unit of length - " nodule».
- We drive a peg at point O and attach the measure “R3 - 3 knots” to it.
- We stretch the rope along the known boundary - towards the proposed point A.
- At the moment of tension on the border line - point A, we drive in a peg.
- Then - again from point O, stretch the measure R4 - along the second border. We don’t drive the peg in yet.
- After this, we stretch the measure R5 - from A to B.
- We drive a peg at the intersection of measurements R2 and R3. – 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 method. Using a compass.
The compass may be rope or pedometer. Cm:
Our compass pedometer has a step of 1 meter.
Ill.2. Compass pedometer
Construction - also according to Ill. 1.
- From the reference point - point O - the neighbor's corner, draw a segment of arbitrary length - but larger than the radius of the compass = 1m - in each direction from the center (segment AB).
- We place the leg of the compass at point O.
- We draw a circle with radius (compass pitch) = 1 m. It is enough to draw short arcs - 10-20 centimeters each, at the intersection with the marked segment (through points A and B). With 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. You can reconfigure our compass to a larger radius if it has an adjustable pitch. But for such a small current task, I wouldn’t want to “pull” it. Or when there is no adjustment. Can be done in half a minute rope compass.
- We place the first nail (or the leg of a compass with a radius greater than 1 m) alternately at points A and B. And draw two arcs with the second nail - in a taut state of the rope - 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 will suffice.
- 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 discrepancy across a neighbor's property. Ill. 3a shows a case where a neighbor’s fence moves away from the desired direction to its detriment. 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 rear boundary of the site. - So that the cord barely touches peg O. By measuring from point O - in direction D, the length of the side according to the general plan, you will get a reliable rear right corner of the site.
Ill.3. Construction right angle– from the neighbor’s corner, using a pedometer and a rope compass
If you have a compass-pedometer, then you can do without rope altogether. In the previous example, we used the rope one to draw arcs of a larger radius than those of 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 with R = 1m.
- Then measure these equidistant points roulette- in different directions from the center, but always along line AB (neighbor’s fence line). The closer points A and B are to the center, the farther the guide points C and D are from it, and the more accurate the measurements. In the figure, this distance is taken to be about a quarter of the pedometer radius = 260mm.
Ill.4. Constructing a right angle using a pedometer and tape measure
- This scheme of actions is no less relevant when constructing any rectangle, in particular the contour of a rectangular foundation. You will receive it perfect. Its diagonals, of course, need to be checked, but isn't the effort reduced? – Compared to when the diagonals, corners and sides of the foundation contour are moved back and forth until the corners meet..
Actually, we solved a geometric problem on earth. To make your actions more confident on the site, practice on paper - using a regular compass. Which is basically no different.
To construct any drawing or perform planar marking of a workpiece before processing it, it is necessary to carry out a number of graphic operations - geometric constructions.
In Fig. Figure 2.1 shows a flat part - a plate. To draw its drawing or mark a contour on a steel strip for subsequent manufacturing, you need to do it on the construction plane, the main ones are numbered with numbers written on the pointer arrows. In numbers 1 indicates the construction of mutually perpendicular lines, which must be performed in several places, with the number 2 – drawing parallel lines, in numbers 3 – pairing these parallel lines with an arc of a certain radius, a number 4 – conjugation of an arc and a straight arc of a given radius, which in this case is 10 mm, number 5 – conjugation of two arcs with an arc of a certain radius.
As a result of performing these and other geometric constructions, the contour of the part will be drawn.
Geometric construction is a method of solving a problem in which the answer is obtained graphically without any calculations. Constructions are carried out with drawing (or marking) tools as carefully as possible, because the accuracy of the solution depends on this.
The lines specified by the conditions of the problem, as well as the constructions, are made solid thin, and the results of the construction are solid main ones.
When starting to make a drawing or marking, you must first determine which of the geometric constructions need to be applied in this case, i.e. analyze the graphic composition of the image.
Rice. 2.1.
Analysis of the graphic composition of the image called the process of dividing the execution of a drawing into separate graphic operations.
Identifying the operations required to construct a drawing makes it easier to choose how to execute it. If you need to draw, for example, the plate shown in Fig. 2.1, then analysis of the contour of its image leads us to the conclusion that we must apply the following geometric constructions: in five cases, draw mutually perpendicular center lines (figure 1 in a circle), in four cases draw parallel lines(number 2 ), draw two concentric circles (0 50 and 70 mm), in six cases construct mates of two parallel straight lines with arcs of a given radius (figure 3 ), and in four - the pairing of an arc and a straight arc of radius 10 mm (figure 4 ), in four cases, construct a pairing of two arcs with an arc of radius 5 mm (number 5 in a circle).
To carry out these constructions, you need to remember or repeat from the textbook the rules for drawing them.
In this case, it is advisable to choose a rational way to complete the drawing. Choosing a rational way to solve a problem reduces the time spent on work. For example, when building equilateral triangle, inscribed in a circle, a more rational method is to construct it using a crossbar and a square with an angle of 60° without first determining the vertices of the triangle (see Fig. 2.2, a, b). A less rational way to solve the same problem is using a compass and a crossbar with preliminary determination of the vertices of the triangle (see Fig. 2.2, V).
Dividing segments and constructing angles
Constructing right angles
It is rational to construct a 90° angle using a crossbar and a square (Fig. 2.2). To do this, it is enough to draw a straight line and restore a perpendicular to it using a square (Fig. 2.2, A). It is rational to build a perpendicular to the inclined segment by moving (Fig. 2.2, b) or turning (Fig. 2.2, V) square.
Rice. 2.2.
Construction of obtuse and acute angles
Rational methods for constructing angles of 120, 30 and 150, 60 and 120, 15 and 165, 75 and 105.45 and 135° are shown in Fig. 2.3, which shows the positions of the squares for constructing these angles.
Rice. 2.3.
Dividing an angle into two equal parts
From the vertex of the corner, describe an arc of a circle of arbitrary radius (Fig. 2.4).
Rice. 2.4.
From points ΜηΝ intersection of an arc with the sides of an angle with a compass solution larger than half the arc ΜΝ, make two intersecting at a point A serifs.
Through the received point A and the vertex of the angle draw a straight line (the bisector of the angle).
Dividing a right angle into three equal parts
From the vertex of a right angle, describe an arc of a circle of arbitrary radius (Fig. 2.5). Without changing the angle of the compass, make notches from the points of intersection of the arc with the sides of the angle. Through the received points M And Ν and the vertex of the angle are drawn by straight lines.
Rice. 2.5.
In this way, only right angles can be divided into three equal parts.
Constructing an angle equal to a given one. From the top ABOUT given angle draw an arc of arbitrary radius R, intersecting the sides of the angle at points M And N(Fig. 2.6, A). Then draw a straight segment, which will serve as one of the sides of the new angle. From point ABOUT 1 on this straight line with the same radius R draw an arc, getting a point Ν 1 (Fig. 2.6, b). From this point describe an arc of radius R 1, equal to the chord MN. The intersection of arcs gives a point Μ 1, which is connected by a straight line to the vertex of the new angle (Fig. 2.6, b).
Rice. 2.6.
Dividing a line segment into two equal parts. Arcs are drawn from the ends of a given segment with a compass opening greater than half its length (Fig. 2.7). Straight line connecting the obtained points M And Ν, divides a segment into two equal parts and is perpendicular to it.
Rice. 2.7.
Constructing a perpendicular at the end of a straight line segment. From an arbitrary point O taken above the segment AB, describe a circle passing through a point A(end of a line segment) and intersecting the line at the point M(Fig. 2.8).
Rice. 2.8.
Through the received point M and center ABOUT circles draw a straight line until they meet opposite side circle at a point N. Full stop N connect a straight line to a point A.
Dividing a line segment into any number of equal parts. From any end of a segment, for example from a point A, draw a straight line at an acute angle to it. On it, using a measuring compass, the required number of equal segments of arbitrary size is laid out (Fig. 2.9). Last point connect to the second end of a given segment (to a point IN). From all division points, using a ruler and a square, draw straight lines parallel to the straight line 9V, which will divide the segment AB into a given number of equal parts.
Rice. 2.9.
In Fig. Figure 2.10 shows how to apply this construction to mark the centers of holes evenly spaced on a straight line.
Lesson objectives:
- Formation of the ability to analyze the studied material and the skills of applying it to solve problems;
- Show the significance of the concepts being studied;
- Development cognitive activity and independence in acquiring knowledge;
- Cultivating interest in the subject and a sense of beauty.
Lesson objectives:
- Develop skills in constructing an angle equal to a given one using a scale ruler, compass, protractor and drawing triangle.
- Test students' problem-solving skills.
Lesson plan:
- Repetition.
- Constructing an angle equal to a given one.
- Analysis.
- Construction example first.
- Construction example two.
Repetition.
Corner.
Flat angle- unlimited geometric figure, formed by two rays (sides of an angle) emerging from one point (vertex of an angle).
An angle is also called a figure formed by all points of the plane enclosed between these rays (Generally speaking, two such rays correspond to two angles, since they divide the plane into two parts. One of these angles is conventionally called internal, and the other - external.
Sometimes, for brevity, the angle is called the angular measure.
There is a generally accepted symbol to denote an angle: , proposed in 1634 by the French mathematician Pierre Erigon.
Corner is a geometric figure (Fig. 1), formed by two rays OA and OB (sides of the angle), emanating from one point O (vertex of the angle).
An angle is denoted by a symbol and three letters indicating the ends of the rays and the vertex of the angle: AOB (and the letter of the vertex is the middle one). Angles are measured by the amount of rotation of ray OA around vertex O until ray OA moves to position OB. There are two widely used units for measuring angles: radians and degrees. For radian measurement of angles, see below in the paragraph “Arc Length”, as well as in the chapter “Trigonometry”.
Degree system for measuring angles.
Here the unit of measurement is a degree (its designation is °) - this is a rotation of the beam by 1/360 of a full revolution. Thus, full turn beam is equal to 360 o. One degree is divided into 60 minutes (symbol ‘); one minute – respectively for 60 seconds (designation “). An angle of 90° (Fig. 2) is called right; an angle less than 90° (Fig. 3) is called acute; an angle greater than 90° (Fig. 4) is called obtuse.
Straight lines forming a right angle are called mutually perpendicular. If the lines AB and MK are perpendicular, then this is denoted: AB MK.
Constructing an angle equal to a given one.
Before starting construction or solving any problem, regardless of the subject, you need to carry out analysis. Understand what the assignment says, read it thoughtfully and slowly. If after the first time you have doubts or something was not clear or clear but not completely, it is recommended to read it again. If you are doing an assignment in class, you can ask the teacher. Otherwise, your task, which you misunderstood, may not be solved correctly, or you may find something that is not what was required of you, and it will be considered incorrect and you will have to redo it. As for me - It’s better to spend a little more time studying the task than to redo the task all over again.
Analysis.
Let a be the given ray with vertex A, and the angle (ab) be the desired one. Let's choose points B and C on rays a and b, respectively. By connecting points B and C, we get triangle ABC. In congruent triangles, the corresponding angles are equal, and this is where the method of construction follows. If on the sides of a given angle we select points C and B in some convenient way, and from a given ray into a given half-plane we construct a triangle AB 1 C 1 equal to ABC (and this can be done if we know all the sides of the triangle), then the problem will be solved.
When carrying out any constructions Be extremely careful and try to carry out all constructions carefully. Since any inconsistencies can result in some kind of errors, deviations, which can lead to an incorrect answer. And if a task of this type is performed for the first time, the error will be very difficult to find and fix.
Construction example first.
Let's draw a circle with its center at the vertex of this angle. Let B and C be the points of intersection of the circle with the sides of the angle. With radius AB we draw a circle with the center at point A 1 – the starting point of this ray. Let us denote the point of intersection of this circle with this ray as B 1 . Let us describe a circle with center at B 1 and radius BC. The intersection point C 1 of the constructed circles in the indicated half-plane lies on the side of the desired angle.
Triangles ABC and A 1 B 1 C 1 are equal on three sides. Angles A and A 1 are the corresponding angles of these triangles. Therefore, ∠CAB = ∠C 1 A 1 B 1
For greater clarity, you can consider the same constructions in more detail.
Construction example two.
The task remains to also set aside an angle equal to a given angle from a given half-line into a given half-plane.
Construction.
Step 1. Let us draw a circle with an arbitrary radius and centers at vertex A of a given angle. Let B and C be the points of intersection of the circle with the sides of the angle. And let's draw segment BC.
Step 2. Let's draw a circle of radius AB with the center at point O - the starting point of this half-line. Let us denote the point of intersection of the circle with the ray as B 1 .
Step 3. Now we describe a circle with center B 1 and radius BC. Let point C 1 be the intersection of the constructed circles in the indicated half-plane. 
Step 4. Let's draw a ray from point O through point C 1. Angle C 1 OB 1 will be the desired one.
Proof.
Triangles ABC and OB 1 C 1 are congruent triangles with corresponding sides. And therefore angles CAB and C 1 OB 1 are equal.
Interesting fact:
In numbers.
In objects of the surrounding world, you first of all notice their individual properties that distinguish one object from another.
The abundance of particular, individual properties obscures the general properties inherent in absolutely all objects, and it is therefore always more difficult to detect such properties.
One of the most important general properties of objects is that all objects can be counted and measured. We reflect this general property objects in the concept of number.
People mastered the process of counting, that is, the concept of number, very slowly, over centuries, in a persistent struggle for their existence.
In order to count, one must not only have objects that can be counted, but also already have the ability to abstract when considering these objects from all their other properties except number, and this ability is the result of a long historical development based on experience.
Every person now learns to count with the help of numbers imperceptibly in childhood, almost simultaneously with the time he begins to speak, but this counting, which is familiar to us, has gone through a long path of development and has taken different forms.
There was a time when only two numerals were used to count objects: one and two. In the process of further expansion of the number system, parts were involved human body and first of all, fingers, and if this kind of “numbers” was not enough, then also sticks, stones and other things.
N. N. Miklouho-Maclay in his book "Trips" talks about a funny method of counting used by the natives of New Guinea:
Questions:
- Define angle?
- What types of angles are there?
- What is the difference between diameter and radius?
List of sources used:
- Mazur K. I. “Solving the main competition problems in mathematics of the collection edited by M. I. Skanavi”
- Mathematical savvy. B.A. Kordemsky. Moscow.
- L. S. Atanasyan, V. F. Butuzov, S. B. Kadomtsev, E. G. Poznyak, I. I. Yudina “Geometry, 7 – 9: textbook for educational institutions”
Worked on the lesson:
Levchenko V.S.
Poturnak S.A.
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