The mathematical concept of a straight line ray segment. The simplest geometric figures: point, straight line, segment, ray, broken line

28.09.2019

A point and a straight line are the basic geometric figures on a plane.

The ancient Greek scientist Euclid said: “a point” is something that has no parts.” The word “point” translated from Latin means the result of an instant touch, an injection. A point is the basis for constructing any geometric figure.

A straight line or simply a straight line is a line along which the distance between two points is the shortest. A straight line is infinite, and it is impossible to depict the entire straight line and measure it.

Points are denoted by capital Latin letters A, B, C, D, E, etc., and straight lines by the same letters, but lowercase a, b, c, d, e, etc. A straight line can also be denoted by two letters corresponding to points lying on her. For example, straight line a can be designated AB.

We can say that points AB lie on line a or belong to line a. And we can say that straight line a passes through points A and B.

Protozoa geometric figures on a plane it is a segment, a ray, broken line.

A segment is a part of a line that consists of all points of this line, limited by two selected points. These points are the ends of the segment. A segment is indicated by indicating its ends.

A ray or half-line is a part of a line that consists of all points of this line lying on one side of a given point. This point is called the starting point of the half-line or the beginning of the ray. The beam has a starting point, but no end.

Half-lines or rays are designated by two lowercase Latin letters: the initial and any other letter corresponding to a point belonging to the half-line. In this case, the starting point is placed in the first place.

It turns out that the straight line is infinite: it has neither beginning nor end; a ray has only a beginning, but no end, but a segment has a beginning and an end. Therefore, we can only measure a segment.

Several segments that are sequentially connected to each other so that the segments (neighboring) that have one common point are not located on the same straight line represent a broken line.

A broken line can be closed or open. If the end of the last segment coincides with the beginning of the first, we have a closed broken line; if not, it is an open line.

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During the lesson you will become familiar with the concept of a plane, with various minimal figures that exist in geometry, and study their properties. Learn what a straight line, segment, ray, angle, etc. are.

We depict all geometric figures on a sheet of paper with a pencil, on school board chalk or marker. Often in the summer we draw figures on the asphalt with chalk or a white pebble. And always, before we start drawing what we have planned, we evaluate whether we have enough space. And since we rarely know the exact dimensions of our future drawing, we always need to take space with a margin, and preferably with a large margin. Usually we are not afraid of running out of space to draw if the field to draw is many times larger than the drawing itself. So there is enough asphalt in the yard to create a jumping field. Notebook sheet enough to draw two intersecting segments in the middle.

In mathematics, the field on which we depict everything is a plane (Fig. 1).

Rice. 1. Plane

She has two qualities:

1. You can depict any figure on it that we have already talked about, or will talk about again.

2. We won't reach the edge. Its dimensions can be considered much larger than the dimensions of the picture.

The fact that we never reach the edge of the plane can be understood as the absence of edges at all. We don’t need its edges, so we agreed to assume that they don’t exist (Fig. 2).

Rice. 2. The plane is infinite

In this sense, the plane is infinite in any direction.

We can think of it as large leaf paper, a large flat asphalt area or a huge drawing board.

There are an infinite number of geometric shapes, and it is absolutely impossible to study them all. But geometry works much like a construction set. There are several types of basic parts from which you can build everything else, any most complex building.

This principle can be compared to words and letters: we know all the letters, but we do not know all the words. When we encounter an unfamiliar word, we can read it because we know how the letters are written and how the corresponding sounds are pronounced.

It’s the same in mathematics - there are very few basic geometric figures that you and I need to know well.

Let's consider a segment (Fig. 3). A segment is shortest line, connecting two points.

Rice. 3. Segment

Let's continue the segment in both directions to infinity. We will also continue straight ahead.

What does "straight" mean? Let's consider the segments and (Fig. 4).

Rice. 4. Segments and

Let's continue them in both directions. The top line is straight, but the bottom line is not (Fig. 5).

Let's add one more point to the top and bottom lines (Fig. 6). The part of the upper line between the points and is also a segment, but the part of the bottom line between the points and the segment is not, since it does not connect these points along the shortest path.

Rice. 6. Continuation of lines and

A straight line is a line that continues indefinitely in both directions, any part of which, limited by two points, is a segment.

A straight line is a type of line, and like any line, a straight line is a figure. And, as for any line, a given point either belongs to a given line or not (Fig. 7).

Rice. 7. Points and belonging to a line, and points and not belonging to a line

1. A straight line divides the plane into two parts, into two half-planes. In Figure 8, the points and lie in the same half-plane, and and - in different half-planes.

Rice. 8. Two half-planes

2. You can always draw a straight line through two points, and only one (Fig. 9).

A straight line, like any line, can be marked with one lowercase letter of the Latin alphabet or a sequence of points that lie on it. To designate a line through the points lying on it, two points are enough.

Extending the segment in both directions to infinity, we got a straight line. If we also extend the segment, but only in one direction to infinity, we get a figure called a ray (Fig. 10). This geometric beam is very similar to a light beam, which is why it is called that. If you pick up a laser pointer, the beam of light will start at the pointer and go to infinity in a straight line.

Rice. 10. Beam

The point is called the beginning of the ray. The ray is indicated.

If you mark a point on a straight line, then it divides this straight line into two rays (Fig. 11). Both rays originate at point , but are directed in different directions. These two rays make up a straight line and are its halves. Therefore, the beam is often also called “half-direct”.

Rice. 11. A point divides a line into two rays

Consider Figure 12.

Rice. 12. Segment, straight line and ray

Let’s figure out how a segment, a straight line and a ray are similar and dissimilar to each other:

The segment and the beam can easily be completed to a straight line; for this, the segment needs to be extended in both directions, and the beam in one direction;

You can always select a segment or ray on a straight line;

The point divides the line into two rays, into two half-lines;

Points and limit to a straight segment;

All these figures: a segment, a ray, a straight line are “straight lines”. They differ in the presence of ends. A segment has two, a ray has one, a straight line has none. Another way to put it is this: both the ray and the segment are part of a straight line;

We know that a segment can have its length measured. Two segments can be compared to find out which one is longer;

The straight line continues indefinitely in both directions, the ray continues in one direction. For this reason, it is impossible to measure the length of a straight line or beam, and it is also impossible to compare the length of two straight lines or two beams. They are all equally infinite.

Two rays, having their origins at the same point, form another geometric figure from the main set - an angle. The point at the beginning of both rays is called the vertex of the angle. The rays themselves are called the sides of the angle.

So, an angle is a figure consisting of two rays emerging from one point (Fig. 13).

Rice. 13. Angle

The angle is designated by one letter corresponding to the designation of the vertex. IN in this case the angle can be called an angle (Fig. 14). To make it clear that we are talking about an angle, and not about a point, before its name you need to write the word “angle” or put a special angle sign (“”).

Rice. 14. Angle

If it is difficult to understand from the vertex which angle we are talking about, as in Figure 15, then use two more points on both sides of the angle.

If we simply name the angle in this figure, it is not clear which one we are talking about, because with the vertex at a point we see several angles. Therefore, we will add a point to the sides of the angle we need and denote the angle as (Fig. 15).

Rice. 15. Angle

When designating, you can go in the opposite direction, but so that the vertex again ends up in the middle of the notation.

Another common designation is with one Greek letter: alpha, beta, gamma, and so on (Fig. 16). In this case, the letter is usually written inside the corner (Fig. 17).

Rice. 16. Greek alphabet

Rice. 17. The name of the angle written inside the angle

So, in Figure 18, the designations , , are equivalent and denote the same angle.

Rice. 18... - same angle

Let two straight lines intersect at a point (Fig. 19). The point divides each line into two rays, that is, 4 rays in total. Each pair of rays sets an angle.

Rice. 19. Straight and form 4 beams

For example, , , .

Through two points you can always draw a straight line. Is this the case with three dots?

In Figure 20 you can draw a straight line through three points, but in Figure 21 you cannot.

Rice. 20. Through three points you can draw a straight line

Rice. 21. You cannot draw a straight line through three points

Three points in the figure are said to lie on the same straight line. This is said even if the straight line itself is not drawn, simply implying that it can be drawn. In the second case, they say that the points do not lie on the same line, implying that it is impossible to draw a line through all three points.

If we connect sequentially first the 1st and 2nd points, then the 2nd and 3rd, then the resulting line is called a broken line (Fig. 22). The name follows from its appearance.

Rice. 22. Broken

Similar to a polyline, you can connect any number of points. The points , , , , are called the vertices of the broken line, the segments , , , are called the links of the broken line.

A broken line is indicated by its vertices.

Rice. 23. Broken

If the last point is connected to the first, then the resulting broken line is called closed (Fig. 24).

Rice. 24. Closed polyline

What kind of polyline can be constructed with minimum set vertices and links? If there are two points, then they can be connected by a segment. This will be the most simple example broken line: two vertices and one link connecting them. We can say that a segment is a minimal broken line.

If it is required that the broken line be closed, then the simplest such broken line will be a triangle. If you take two points, then you can connect the last point with the first only with the same segment that already exists. That is, the broken line will remain, as before, open. And if you add one more point that does not lie on the same straight line with the points and , connect all the points with three segments, you get a triangle (Fig. 25).

Rice. 25. Triangle

A triangle is a closed broken line with three vertices. Or even like this: a triangle is a minimal closed broken line.

Points , and are the vertices of the triangle. The segments connecting them, the links of the broken line, are called the sides of the triangle.

A triangle is designated by its vertices. For example, . Before the designation you need to put the word “triangle” or a special triangle symbol (“”).

A triangle implies three angles. Two sides emanate from each of the vertices, that is, the sides of the triangle are the sides of the angles (Fig. 26).

Rice. 26. Angles of a triangle

Thus, a triangle has three vertices (three points, and), three sides (three segments, and).

A point and a straight line are the basic geometric figures on a plane.

We can say that points AB lie on line a or belong to line a. And we can say that straight line a passes through points A and B.

The simplest geometric figures on a plane are a segment, a ray, a broken line.

A segment is a part of a line that consists of all points of this line, limited by two selected points. These points are the ends of the segment. A segment is indicated by indicating its ends.

Several segments that are sequentially connected to each other so that the segments (neighboring) that have one common point are not located on the same straight line represent a broken line.

A broken line can be closed or open. If the end of the last segment coincides with the beginning of the first, we have a closed broken line; if not, it is an open line.

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A point is an abstract object that has no measuring characteristics: no height, no length, no radius. Within the scope of the task, only its location is important

The point is indicated by a number or a capital (capital) Latin letter. Several dots - different numbers or in different letters so that they can be distinguished

point A, point B, point C

A B C

point 1, point 2, point 3

1 2 3

You can draw three dots “A” on a piece of paper and invite the child to draw a line through the two dots “A”. But how to understand through which ones? A A A

A line is a set of points. Only the length is measured. It has no width or thickness

Indicated by lowercase (small) Latin letters

line a, line b, line c

a b c

The line may be

closed if its beginning and end are at the same point,
open if its beginning and end are not connected

closed lines

open lines

You left the apartment, bought bread at the store and returned back to the apartment. What line did you get? That's right, closed. You are back to your starting point. You left the apartment, bought bread at the store, went into the entrance and started talking with your neighbor. What line did you get? Open. You haven't returned to your starting point. You left the apartment and bought bread at the store. What line did you get? Open. You haven't returned to your starting point.

self-intersecting
without self-intersections

self-intersecting lines

lines without self-intersections

straight
broken
crooked

straight lines

broken lines

curved lines

A straight line is a line that is not curved, has neither beginning nor end, it can be continued endlessly in both directions

Even when a small section of a straight line is visible, it is assumed that it continues indefinitely in both directions

Indicated by a lowercase (small) Latin letter. Or two capital (capital) Latin letters - points lying on a straight line

straight line a

straight line AB

B A

Direct may be

intersecting if they have a common point. Two lines can intersect only at one point.
- perpendicular if they intersect at right angles (90°).
Parallel, if they do not intersect, do not have a common point.

parallel lines

intersecting lines

perpendicular lines

A ray is a part of a straight line that has a beginning but no end; it can be continued indefinitely in only one direction

The ray of light in the picture has its starting point as the sun.

Sun

A point divides a straight line into two parts - two rays A A

The beam is designated by a lowercase (small) Latin letter. Or two capital (capital) Latin letters, where the first is the point from which the ray begins, and the second is the point lying on the ray

ray a

beam AB

B A

The rays coincide if

located on the same straight line
start at one point
directed in one direction

rays AB and AC coincide

rays CB and CA coincide

C B A

A segment is a part of a line that is limited by two points, that is, it has both a beginning and an end, which means its length can be measured. The length of a segment is the distance between its starting and ending points

Through one point you can draw any number of lines, including straight lines

Through two points - an unlimited number of curves, but only one straight line

curved lines passing through two points

B A

straight line AB

B A

A piece was “cut off” from the straight line and a segment remained. From the example above you can see that its length is the shortest distance between two points. ✂ B A ✂

A segment is denoted by two capital (capital) Latin letters, where the first is the point at which the segment begins, and the second is the point at which the segment ends

segment AB

B A

Problem: where is the line, ray, segment, curve?

A broken line is a line consisting of consecutively connected segments not at an angle of 180°

A long segment was “broken” into several short ones

The links of a broken line (similar to the links of a chain) are the segments that make up the broken line. Adjacent links are links in which the end of one link is the beginning of another. Adjacent links should not lie on the same straight line.

The vertices of a broken line (similar to the tops of mountains) are the point from which the broken line begins, the points at which the segments that form the broken line are connected, and the point at which the broken line ends.

A broken line is designated by listing all its vertices.

broken line ABCDE

vertex of polyline A, vertex of polyline B, vertex of polyline C, vertex of polyline D, vertex of polyline E

broken link AB, broken link BC, broken link CD, broken link DE

link AB and link BC are adjacent

link BC and link CD are adjacent

link CD and link DE are adjacent

A B C D E 64 62 127 52

The length of a broken line is the sum of the lengths of its links: ABCDE = AB + BC + CD + DE = 64 + 62 + 127 + 52 = 305

Task: which broken line is longer, A which has more vertices? The first line has all the links of the same length, namely 13 cm. The second line has all links of the same length, namely 49 cm. The third line has all the links of the same length, namely 41 cm.

A polygon is a closed polyline

The sides of the polygon (the expressions will help you remember: “go in all four directions”, “run towards the house”, “which side of the table will you sit on?”) are the links of a broken line. Adjacent sides of a polygon are adjacent links of a broken line.

The vertices of a polygon are the vertices of a broken line. Adjacent vertices are the endpoints of one side of the polygon.

A polygon is denoted by listing all its vertices.

closed polyline without self-intersection, ABCDEF

polygon ABCDEF

polygon vertex A, polygon vertex B, polygon vertex C, polygon vertex D, polygon vertex E, polygon vertex F

vertex A and vertex B are adjacent

vertex B and vertex C are adjacent

vertex C and vertex D are adjacent

vertex D and vertex E are adjacent

vertex E and vertex F are adjacent

vertex F and vertex A are adjacent

polygon side AB, polygon side BC, polygon side CD, polygon side DE, polygon side EF

side AB and side BC are adjacent

side BC and side CD are adjacent

CD side and DE side are adjacent

side DE and side EF are adjacent

side EF and side FA are adjacent

A B C D E F 120 60 58 122 98 141

The perimeter of a polygon is the length of the broken line: P = AB + BC + CD + DE + EF + FA = 120 + 60 + 58 + 122 + 98 + 141 = 599

A polygon with three vertices is called a triangle, with four - a quadrilateral, with five - a pentagon, etc.