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.
All geometric shapes we draw on a piece of paper with a pencil, on school blackboard 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 is structured 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 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, and 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 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 what exactly 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 same 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 can’t 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).
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 ray OD and ED
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 on a house wall. 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: 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 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 just 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 on opposite sides of 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 significance 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 a given point on the plane there passes a single 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 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 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.
References.
- 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. Analytical geometry.
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Visiting additional classes we realized that we do not know how to operate with the concepts of point, line, angle, ray, segment, straight, curve, closed line and draw them; more precisely, we can draw, but we cannot identify them.
Children must recognize lines, curves, and circles. This develops their graphics and sense of correctness when practicing drawing and appliqué. It is important to know what basic geometric shapes exist and what they are. Lay out the cards in front of the child and ask them to draw exactly the same as in the picture. Repeat several times.
During the classes we were given the following materials:
A little fairy tale.
In the land of Geometry there lived a dot. She was small. It was left by a pencil when it stepped on a piece of notebook paper, and no one noticed it. So she lived until she came to visit the lines. (There is a drawing on the board.)
Look what those lines were. (Straight and curved.)
Straight lines are like stretched strings, and strings that are not stretched are crooked lines.
How many straight lines? (2.)
How many curves? (3.)
The straight line began to boast: “I am the longest! I have neither beginning nor end! I am endless!
It became very interesting to look at her. The point itself is tiny. She came out and was so carried away that she didn’t notice how she stepped on a straight line. And suddenly the straight line disappeared. A beam appeared in its place.
It was also very long, but still not as long as a straight line. He got a start.
The dot got scared: “What have I done!” She wanted to run away, but as luck would have it she stepped on the beam again.
And in place of the beam a segment appeared. He didn't brag about how big he was, he already had a beginning and an end.
This is how a small dot was able to change the life of large lines.
So who guessed who came to visit us with the cat? (straight line, ray, segment and point)
That's right, along with the cat, a straight line, a ray, a segment and a point came to our lesson.
Who guessed what we will do in this lesson? (Learn to recognize and draw a straight line, ray, segment.)
What lines did you learn about? (About a line, ray, segment.)
What did you learn about the straight line? (It has neither beginning nor end. It is endless.)
(We take two spools of thread, pull them, depicting a straight line, and unwinding first one, then the other, demonstrates that the straight line can be continued in both directions ad infinitum.)
What did you learn about the ray? (It has a beginning, but no end.) (The teacher takes scissors, cuts the thread. Shows that now the line can only be continued in one direction.)
What did you learn about the segment? (It has both a beginning and an end.) (The teacher cuts off the other end of the thread and shows that the thread does not stretch. It has both a beginning and an end.)
How to draw a straight line? (Draw a line along the ruler.)
How to draw a line segment? (Put two points and connect them.)
And of course the copybook:
