Designations and symbolism. The simplest geometric figures: point, straight line, segment, ray, broken line

In geometry, the main geometric figures are the point and the line. To designate points, it is customary to use capital Latin letters: A, B, C, D, E, F.... To designate straight lines, lowercase Latin letters are used: a, b, c, d, e, f .... The figure below shows straight line a, and several points A, B, C, D.

To depict a straight line in the drawing, we use a ruler, but we do not depict the entire straight line, but only a piece of it. Since the straight line in our representation extends to infinity in both directions, the straight line is infinite.

In the figure presented above we see that points A and C are located on a straight line A. In such cases, they say that points A and C belong to line a. Or they say that a straight line passes through points A and C. When writing, the belonging of a point to a straight line is indicated by a special icon. And the fact that the point does not belong to the line is marked with the same icon, only crossed out.

In our case, points B and D do not belong to line a.

As noted above, in the figure points A and C belong to straight line a. The part of a line that consists of all the points of this line lying between two given points is called segment. In other words, a segment is a part of a line bounded by two points.

In our case we have a segment AB. Points A and B are called the ends of the segment. In order to designate a segment, its ends are indicated, in our case AB. One of the main properties of belonging of points and lines is the following property: through any two points you can draw a straight line, and only one.

If two lines have common point, then we say that these two lines intersect. In the figure, lines a and b intersect at point A. Lines a and c do not intersect.

Any two lines have only one common point or no common points. If we assume the opposite, that two lines have two points in common, then two lines would pass through them. But this is impossible, since only one straight line can be drawn through two points.

Despite the fact that geometry is one of the exact sciences, scientists cannot unambiguously define the term “straight line”. In the very general view we can give the following definition: “A straight line is a line along which the path is equal to the distance between two points.”

What is a straight line in mathematics? The definition of a straight line in mathematics is that a straight line has no ends and can continue in both directions indefinitely.

The basic concepts of geometry include point, line and plane; they are given without definition, but definitions of other geometric figures are given through these concepts. A plane, like a straight line, is a primary concept that has no definition. This statement is established by the following axiom: if two points of a line lie in a certain plane, then all points of this line lie in this plane. And the statement itself that is being proven is called a theorem. The formulation of the theorem usually consists of two parts.

Problem: where is the line, ray, segment, curve? 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, the point at which the broken line ends. Problem: which broken line is longer and which has more vertices? 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.

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

In the future there will be definitions for different figures except two - a point and a straight line. This means that sometimes we can denote a straight line with two large with Latin letters, for example, line \(AB\), since no other line can be drawn through these two points. Symbolically we write the segment \(AB\).

What is a point in mathematics?

Theorem: The midline of a triangle is parallel to one of its sides and equal to half of that side. C. Altitude of a right triangle drawn from the vertex right angle, divides a triangle into two similar ones right triangle, each of which is similar to a given triangle. C. An inscribed angle subtended by a semicircle is a right angle. Here are the basic definitions, theorems, and properties of figures on the plane.

The vector with the coordinates of the point is called a normal vector; it is perpendicular to the line.

In a systematic presentation of geometry, a straight line is usually taken as one of the initial concepts, which is only indirectly determined by the axioms of geometry.

4. Two divergent lines on a plane either intersect at a single point, or they are parallel. A ray is a part of a straight line limited on one side. A segment, like a straight line, is denoted by either one letter or two. In the latter case, these letters indicate the ends of the segment.

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

Point in mathematics

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

In the figure you can see an 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 the set of all points lying on a straight line between the ends of the 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.

Direct in mathematics

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

The figure shows two straight lines: CD and EF.

Beam 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 starting point of the beam, so letters cannot be swapped.

The figure shows the rays: DC, KC, EF, MT, MS. Beams KC and KD are 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 the number line, as well as the OD and ED rays

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§1. Control questions
Question 1. Give examples of geometric shapes.
Answer. Examples of geometric shapes: triangle, square, circle.

Question 2. Name the main ones geometric figures on surface.
Answer. The main geometric figures on a plane are a point and a straight line.

Question 3. How are points and lines designated?
Answer. Points are designated in capital Latin letters: A, B, C, D, …. Direct lines are designated by lowercase Latin letters: a, b, c, d, ….
A straight line can be denoted by two points lying on it. For example, line a in Figure 4 can be labeled AC, and line b can be labeled BC.

Question 4. Formulate the basic properties of membership of points and lines.
Answer. Whatever the line, there are points that belong to this line and points that do not belong to it.
Through any two points you can draw a straight line, and only one.
Question 5. Explain what a line segment with ends at these points is.
Answer. A segment is a part of a line that consists of all points of this line lying between two given points. These points are called the ends of the segment. A segment is indicated by indicating its ends. When they say or write: “segment AB,” they mean a segment with ends at points A and B.

Question 6. State the basic property of the location of points on a straight line.
Answer. From three points On a straight line, one and only one lies between the other two.
Question 7. Formulate the basic properties of measuring segments.
Answer. Each segment has a certain length greater than zero. The length of a segment is equal to the sum of the lengths of the parts into which it is divided by any of its points.
Question 8. What is the distance between two given points?
Answer. The length of segment AB is called the distance between points A and B.
Question 9. What properties does the division of a plane into two half-planes have?
Answer. Partitioning a plane into two half-planes has the following property. If the ends of a segment belong to the same half-plane, then the segment does not intersect the line. If the ends of a segment belong to different half-planes, then the segment intersects a line.

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

Page navigation.

A straight line on a plane is a concept.

Before giving the concept of a straight line on a plane, you should clearly understand what a plane is. Concept of a plane allows you to get, for example, a flat surface on a table or a wall at home. It should, however, 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 its tip to the surface of the “table”, we will get an image of a point. This is how we get representation of a point on a plane.

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

Place a sheet of clean paper on the table surface (on a plane). In order to draw a straight line, we need to take a ruler and draw a line with a pencil as far as the size of the ruler and sheet of paper we are using allows us to do. It should be noted that in this way we will get only part of the line. We can only imagine an entire straight line extending into infinity.

The relative position of a line and a point.

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

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

Possible two options for the relative position of a line and a point on a plane: either the point lies on the line (in this case it is also said that the line passes 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, use the symbol “”. For example, if point A lies on line a, then we can write . If point A does not belong to line a, then write .

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

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 defined on a plane there are infinitely many different points, and all these points lie in the same plane. This statement is established by the axiom: if two points of a line lie in a certain plane, then all points of this line lie in this plane.

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

To briefly record whether a certain point belongs or does not belong to a certain segment, the same symbols and are used. To show that a certain segment lies or does not lie on a line, use the symbols and, respectively. For example, if segment AB belongs to line a, you can briefly write .

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 line, and point B lies between points A and C. Then we can say that points A and C are located along different sides from point B. We 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 on a line divides this 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.

The relative position of lines on a plane.

Now let’s answer the question: “How can two straight lines be located on a plane relative to each other?”

Firstly, two straight lines on a plane can coincide.

This is possible when the lines have at least two common points. Indeed, by virtue of the axiom stated in the previous paragraph, there is only one straight line passing through two points. In other words, if two straight lines pass through two given points, then they coincide.

Secondly, two straight lines on 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 entry means that lines a and b intersect at point M. Intersecting lines lead us to the concept of 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.

Thirdly, two straight lines on 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. Special meaning have non-zero vectors lying on a given line or on any of the parallel lines, they are called directing vectors of a 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 this one. Such vectors are called normal line vectors. The use of normal line vectors is described in the article normal line vector 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 can be parallel, otherwise some or all of them intersect. In this case, all lines can intersect at a single point (see the article on a bunch of lines), or they can have different points of intersection.

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

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 a certain line on a plane intersects one of two parallel lines, then it also intersects the second line.

Methods for defining a straight line on a plane.

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

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

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

If the coordinates of two divergent 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.

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

If a straight line on a plane is specified in this way relative to the entered rectangular Cartesian system coordinates, that is, the ability to create its equation. This is written about in the article equation of a line passing through a given point parallel to a given line.

Thirdly, a straight line can be specified 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 construct 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 indicate the point through which it passes and the line to which it is perpendicular. Indeed, through a given point of the plane there passes a single straight line perpendicular to the given straight line. Let's leave this fact without proof.

Finally, a line in a 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: textbook for general education institutions.
Atanasyan L.S., Butuzov V.F., Kadomtsev S.B., Kiseleva L.S., Poznyak E.G. Geometry. Textbook for 10-11 grades of secondary school.
Bugrov Ya.S., Nikolsky S.M. Higher mathematics. Volume one: elements of linear algebra and analytical geometry.
Ilyin V.A., Poznyak E.G. Analytic geometry.

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