Point straight line segment broken line. Point, line, straight line, ray, segment, broken line

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

The point is indicated by a number or a capital (large) Latin letter. Several dots - different numbers or 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 "A" points on a piece of paper and invite the child to draw a line through the two "A" points. But how to understand through which? A A A

A line is a set of points. She only measures length. It has no width or thickness.

Indicated by lowercase (small) with Latin letters

line a, line b, line c

a b c

The line could be

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

closed lines

open lines

You left the apartment, bought bread in the store and returned back to the apartment. What line did you get? That's right, closed. You have returned to the starting point. You left the apartment, bought bread in the store, went into the entrance and talked to your neighbor. What line did you get? Open. You have not returned to the starting point. You left the apartment, bought bread in the store. What line did you get? Open. You have not returned to the starting point.

1. self-intersecting
2. without self-intersections

self-intersecting lines

lines without self-intersections

1. straight
2. broken line
3. crooked

straight lines

broken lines

curved lines

A straight line is a line that does not curve, has neither beginning nor end, it can be extended indefinitely in both directions.

Even when seen small plot straight, it is assumed that it continues indefinitely in both directions

It is denoted by a lowercase (small) Latin letter. Or two capital (large) Latin letters - points lying on a straight line

straight line a

a

straight line AB

B A

straight lines can be

1. intersecting if they have a common point. Two lines can only intersect at one point.
   • perpendicular if they intersect at a right angle (90°).
2. parallel, if they do not intersect, they 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 extended indefinitely in only one direction

The starting point for the beam of light in the picture is the sun.

Sun

The point divides the line into two parts - two rays A A

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

beam a

a

beam AB

B A

The beams match if

1. located on the same straight line
2. start at one point
3. directed to one side

rays AB and AC coincide

rays CB and CA coincide

C B A

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

Any number of lines can be drawn through one point, including straight lines.

Through two points - 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 (large) Latin letters, where the first is the point from which the segment begins, and the second is the point from which the segment ends

segment AB

B A

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

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

A long segment was “broken” into several short ones.

The links of a polyline (similar to the links of a chain) are the segments that make up the polyline. 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 tops of the polyline (similar to the tops of mountains) are the point from which the polyline begins, the points at which the segments forming the polyline are connected, the point where the polyline ends.

A polyline is denoted 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

link of broken line AB, link of broken line BC, link of broken line CD, link of broken line 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 polyline is the sum of the lengths of its links: ABCDE = AB + BC + CD + DE = 64 + 62 + 127 + 52 = 305

A task: which broken line is longer, but which one has more peaks? At the first line, all the links are of the same length, namely 13 cm. The second line has all the 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 (they will help you remember the expressions: "go to all four sides", "run towards the house", "which side of the table will you sit on?") are the links of the broken line. Adjacent sides of a polygon are adjacent links of a broken line.

The vertices of the polygon are the vertices of the polyline. Neighboring vertices are 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

side CD and side DE 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 polyline: 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, and so on.

In this article, we will dwell in detail on one of the primary concepts of geometry - on the concept of a straight line on a plane. First, let's define the basic terms and notation. Next, we discuss the relative position of a line and a point, as well as two lines on a plane, and give the necessary axioms. In conclusion, we will consider ways to set a straight line on a plane and give graphic illustrations.

Page navigation.

A straight line on a plane is a concept.

Before giving the concept of a straight line on a plane, one should clearly understand what a plane is. Representation of the plane allows you to get, for example, a flat surface of the table or the wall of the house. However, it should be borne in mind that the dimensions of the table are limited, and the plane extends beyond these boundaries to infinity (as if we had an arbitrarily large table).

If we take a well-sharpened pencil and touch the surface of the “table” with its core, then we will get an image of a point. So we get representation of a point on a plane.

Now you can go to concept of a straight line on a plane.

Let's put on the surface of the table (on the plane) a sheet of clean paper. In order to draw a straight line, we need to take a ruler and draw a line with a pencil as far as the dimensions of the ruler and sheet of paper used allow. It should be noted that in this way we get only a part of the straight line. A straight line in its entirety, extending to infinity, we can only imagine.

Mutual position of a line and a point.

You should start with an axiom: there are points on every straight line and in every plane.

Points are usually denoted by capital Latin letters, for example, points A and F. In turn, straight lines are denoted by small Latin letters, for example, straight lines a and d.

Possible two options relative position line and points on the plane: either the point lies on the line (in this case, the line is also said to pass through the point), or the point does not lie on the line (it is also said that the point does not belong to the line, or the line does not pass through the point).

To indicate that a point belongs to a certain line, the symbol "" is used. For example, if point A lies on the line a, then you can write. If point A does not belong to the line a, then write down.

The following statement is true: through any two points there is only one straight line.

This statement is an axiom and should be accepted as a fact. In addition, this is quite obvious: we mark two points on paper, apply a ruler to them and draw a straight line. A straight line passing through two given points (for example, through points A and B), can be denoted by these two letters (in our case, straight line AB or BA).

It should be understood that on a straight line given on a plane, there are infinitely many different points, and all these points lie in the same plane. This assertion is established by the axiom: if two points of a line lie in some plane, then all points of this line lie in this plane.

The set of all points located between two points given on a straight line, together with these points, is called straight line or simply segment. The points that bound the segment are called the ends of the segment. A segment is denoted by two letters corresponding to the points of the ends of the segment. For example, let points A and B be the ends of a segment, then this segment can be denoted AB or BA. Please note that this designation of a segment is the same as the designation of a straight line. To avoid confusion, we recommend adding the word "segment" or "straight" to the designation.

For a short record of belonging and not belonging to a certain point to a certain segment, all the same symbols and are used. To show that a segment lies or does not lie on a straight line, the symbols and are used, respectively. For example, if the segment AB belongs to the line a, you can briefly write down.

We should also dwell on the case when three different points belong to the same line. In this case, one, and only one point, lies between the other two. This statement is another axiom. Let points A, B and C lie on the same straight line, and point B lies between points A and C. Then we can say that points A and C are on opposite sides of point B. You can also say that points B and C lie on the same side of point A, and points A and B lie on the same side of point C.

To complete the picture, we note that any point of a straight line divides this straight line into two parts - two beam. For this case, an axiom is given: an arbitrary point O, belonging to a line, divides this line into two rays, and any two points of one ray lie on the same side of the point O, and any two points of different rays lie on opposite sides of the point O.

Mutual arrangement of straight lines on a plane.

Now let's answer the question: "How can two lines be located on a plane relative to each other"?

First, two lines in a plane can coincide.

This is possible when the lines have at least two points in common. Indeed, by virtue of the axiom voiced in the previous paragraph, a single straight line passes through two points. In other words, if two lines pass through two given points, then they coincide.

Secondly, two straight lines in a plane can cross.

In this case, the lines have one common point, which is called the point of intersection of the lines. The intersection of lines is denoted by the symbol "", for example, the record means that lines a and b intersect at point M. Intersecting lines lead us to the concept of the angle between intersecting lines. Separately, it is worth considering the location of straight lines on a plane when the angle between them is ninety degrees. In this case, the lines are called perpendicular(we recommend the article perpendicular lines, perpendicularity of lines). If line a is perpendicular to line b, then short notation can be used.

Third, two lines in a plane can be parallel.

From a practical point of view, it is convenient to consider a straight line on a plane together with vectors. Of particular importance are non-zero vectors lying on a given line or on any of the parallel lines, they are called direction vectors of the straight line. The article directing vector of a straight line on a plane gives examples of directing vectors and shows options for their use in solving problems.

You should also pay attention to non-zero vectors lying on any of the lines perpendicular to the given one. Such vectors are called normal vectors of the line. The use of normal vectors of a straight line is described in the article normal vector of a straight line on a plane.

When three or more straight lines are given on a plane, then a set arises various options their relative position. All lines may be parallel, otherwise some or all of them intersect. In this case, all lines can intersect at a single point (see the article a pencil of lines), or they can have various points intersections.

We will not dwell on this in detail, but we will cite several remarkable and very often used facts without proof:

• if two lines are parallel to a third line, then they are parallel to each other;
• if two lines are perpendicular to a third line, then they are parallel to each other;
• if in a plane a line intersects one of two parallel lines, then it also intersects the second line.

Methods for setting a straight line on a plane.

Now we will list the main ways in which you can define a specific line in the plane. This knowledge is very useful from a practical point of view, since the solution of so many examples and problems is based on it.

First, a straight line can be defined by specifying two points on the plane.

Indeed, from the axiom considered in the first paragraph of this article, we know that a straight line passes through two points, and moreover, only one.

If the coordinates of two non-coincident points are indicated in a rectangular coordinate system on a plane, then it is possible to write down the equation of a straight line passing through two given points.

Second, a line can be specified by specifying the point through which it passes and the line to which it is parallel. This method is valid, since a single straight line passes through a given point of the plane, parallel to a given straight line. The proof of this fact was carried out at geometry lessons in high school.

If a straight line on a plane is set in this way with respect to the introduced rectangular Cartesian coordinate system, then it is possible to compose its equation. This is written in the article the equation of a straight line passing through a given point parallel to a given straight line.

Thirdly, a line can be defined by specifying the point through which it passes and its direction vector.

If a straight line is given in a rectangular coordinate system in this way, then it is easy to compose its canonical equation of a straight line on a plane and parametric equations of a straight line on a plane.

The fourth way to specify a line is to specify the point through which it passes and the line to which it is perpendicular. Indeed, through given point There is only one line in the plane that is perpendicular to the given line. Let's leave this fact without proof.

Finally, a line in the plane can be specified by specifying the point through which it passes and the normal vector of the line.

If the coordinates of a point lying on a given line and the coordinates of the normal vector of the line are known, then it is possible to write down the general equation of the line.

Bibliography.

• Atanasyan L.S., Butuzov V.F., Kadomtsev S.B., Poznyak E.G., Yudina I.I. Geometry. Grades 7 - 9: a textbook for educational institutions.
• Atanasyan L.S., Butuzov V.F., Kadomtsev S.B., Kiseleva L.S., Poznyak E.G. Geometry. Textbook for 10-11 grades of high school.
• Bugrov Ya.S., Nikolsky S.M. higher mathematics. Volume One: Elements of Linear Algebra and Analytic Geometry.
• Ilyin V.A., Poznyak E.G. Analytic geometry.

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We will look at each of the topics, and at the end there will be tests on the topics.

Point in math

What is a point in mathematics? A mathematical point has no dimensions and is indicated by capital Latin letters: A, B, C, D, F, etc.

In the figure, you can see the image of points A, B, C, D, F, E, M, T, S.

Segment in mathematics

What is a segment in mathematics? In mathematics lessons, you can hear the following explanation: a mathematical segment has a length and ends. A segment in mathematics is a set of all points lying on a straight line between the ends of a segment. The ends of the segment are two boundary points.

In the figure we see the following: segments ,,,, and , as well as two points B and S.

Straight lines in mathematics

What is a straight line in mathematics? Definition of a straight line in mathematics: a straight line has no ends and can continue in both directions to infinity. A straight line in mathematics is denoted by any two points on a straight line. To explain the concept of a straight line to a student, we can say that a straight line is a segment that does not have two ends.

The figure shows two straight lines: CD and EF.

Ray in mathematics

What is a ray? Definition of a ray in mathematics: A ray is a part of a line that has a beginning and no end. The name of the beam contains two letters, for example, DC. Moreover, the first letter always indicates the point of the beginning of the beam, so you cannot swap the letters.

The figure shows the beams: DC, KC, EF, MT, MS. Beams KC and KD - one beam, because they have a common origin.

Number line in mathematics

Definition of a number line in mathematics: A line whose points mark numbers is called a number line.

The figure shows a number line, as well as a ray OD and ED