The simplest geometric shapes: point, straight line, segment, ray, broken line. Lesson "Direct"

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

Question 2. What are the main geometric figures on surface.
Answer. The main geometric figures on the plane are the point and the line.

Question 3. How are points and lines defined?
Answer. Points are indicated by capital letters. with Latin letters: A, B, C, D, … . Straight lines are denoted by lowercase Latin letters: a, b, c, d, ....
A line can be denoted by two points lying on it. For example, line a in figure 4 could be labeled AC, and line b could 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 line, and only one.
Question 5. Explain what a segment with ends at given points is.
Answer. A segment is a part of a straight line that consists of all points of this straight line that lie between two given points of it. 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. Formulate the main property of the location of points on a straight line.
Answer. Of the three points on a line, one and only one lies between the other two.
Question 7. Formulate the main 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 are the properties of splitting a plane into two half-planes?
Answer. The partition of a plane into two half-planes has the following property. If the ends of any segment belong to the same half-plane, then the segment does not intersect the line. If the endpoints of a segment belong to different half-planes, then the segment intersects the line.

Despite the fact that geometry is one of the exact sciences, scientists cannot unambiguously define the term "straight line". In the very general view can be given this 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? Definition of a straight line in mathematics: a straight line has no ends and can continue in both directions to infinity.

The basic concepts of geometry include point, line and plane, they are given without definition, but definitions of other geometric shapes are given through these concepts. A plane, like a straight line, is a primary concept that has no definition. This assertion 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, which is proved, is called a theorem. The statement of the theorem usually consists of two parts.

Task: where is the line, ray, segment, curve? 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. Task: which polyline is longer and which has more vertices? 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.

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.

In the following, there will be definitions for different figures except for two - a point and a line. So sometimes we can designate a straight line with two capital Latin letters, for example, a straight line\(AB\), since no other straight line can be drawn through these two points. We symbolically 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. Height of a right triangle drawn from a vertex right angle, divides the triangle into two similar right triangle, each of which is similar to a given triangle. C. An inscribed angle based on a semicircle is a right angle. Here are collected the main definitions, theorems, properties of figures on the plane.

The vector with the coordinates of the point is called the 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 non-coinciding straight lines in a plane either intersect at a single point, or they are parallel. A ray is a part of a straight line bounded on one side. A segment, like a straight line, is indicated by either one letter or two. In the latter case, these letters indicate the ends of the segment.

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) 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 ray begins, and the second is the point lying on the ray

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

Task: which broken line is longer, a 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 geometry, the main geometric figures are the point and the line. To designate points, it is customary to use uppercase 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 a straight line a, and several points A, B, C, D.

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

In the figure above, we see that points A and C are located on a straight line. a. In such cases, we say that the points A and C belong to the line a. Or they say that the line passes through points A and C. When writing, the belonging of a point to a 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, the points B and D do not belong to the line a.

As noted above, in the figure, points A and C belong to the line a. The part of a line that consists of all points on that line that lie between two given points is called segment. In other words, a segment is a part of a straight 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 membership of points and lines is the following property: through any two points you can draw a line, and moreover, only one.

If two lines have a common point, then the two lines are said to 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 line can be drawn through two points.