How can you designate a straight line with three points? A polygon is a closed broken line. A straight line is a line that is not curved, has neither beginning nor end, it can be continued endlessly in both directions

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.

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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, Smooth surface table or wall of the house. 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 only get 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: on every straight line and in every plane there are points.

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 relative position straight line and points on the 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.

A straight line on a plane with practical point it is convenient to consider 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 defined in this way relative to the introduced rectangular Cartesian coordinate system, then it is possible to compose 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 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 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 given point plane there is only one line perpendicular to the given 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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