"Direct" in geometry. Straight line on a plane - necessary information
Point and line are the main geometric figures on the plane.
The ancient Greek scientist Euclid said: “a point” is that which has no parts.” The word "point" in translation from Latin means the result of an instant touch, a prick. The point is the basis for constructing any geometric figure.
A straight line or just a straight line is a line along which the distance between two points is the shortest. A straight line is infinite, and it is impossible to depict the entire line and measure it.
Points are denoted by capital Latin letters A, B, C, D, E, etc., and straight lines by the same letters, but lowercase a, b, c, d, e, etc. A straight line can also be denoted by two letters corresponding to points lying on her. For example, the line a can be denoted by AB.
We can say that the points AB lie on the line a or belong to the line a. And we can say that the line a passes through the points A and B.
The simplest geometric figures on a plane are a segment, a ray, a broken line.
A segment is a part of a line, which consists of all points of this line, bounded by two selected points. These points are the ends of the segment. A segment is indicated by indicating its ends.
A ray or half-line is a part of a line, which consists of all points of this line, lying on one side of its given point. This point is called the starting point of the half-line or the beginning of the ray. A ray has a start point but no end point.
Half-lines or rays are denoted by two lowercase Latin letters: the initial and any other letter corresponding to a point belonging to the half-line. In this case, the starting point is placed in the first place.
It turns out that the line is infinite: it has neither beginning nor end; a ray has only a beginning but no end, while a segment has a beginning and an end. Therefore, we can only measure a segment.
Several segments that are sequentially connected to each other so that the segments (neighboring) having one common point are not located on the same straight line represent a broken line.
The polyline can be closed or open. If the end of the last segment coincides with the beginning of the first, we have a closed broken line, if not, an open one.
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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 Cpoint 1, point 2, point 3
1 2 3You 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 cThe line could be
- closed if its beginning and end are at the same point,
- 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.- self-intersecting
- without self-intersections
self-intersecting lines
lines without self-intersections
- straight
- broken line
- 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 a small section of a straight line is visible, 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
astraight line AB
B Astraight lines can be
- intersecting if they have a common point. Two lines can only intersect at one point.
- perpendicular if they intersect at a right angle (90°).
- 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
abeam AB
B AThe beams match if
- located on the same straight line
- start at one point
- directed to one side
rays AB and AC coincide
rays CB and CA coincide
C B AA 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 Astraight line AB
B AA 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 ATask: 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 vertices 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 52The 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, 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 141The 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.
Point and line are the main geometric figures on the plane.
The ancient Greek scientist Euclid said: “a point” is that which has no parts.” The word "point" in Latin means the result of an instant touch, a prick. The point is the basis for constructing any geometric figure.
A straight line or just a straight line is a line along which the distance between two points is the shortest. A straight line is infinite, and it is impossible to depict the entire line and measure it.
Points are denoted by capital Latin letters A, B, C, D, E, etc., and straight lines by the same letters, but lowercase a, b, c, d, e, etc. A straight line can also be denoted by two letters corresponding to points lying on her. For example, the line a can be denoted by AB.
We can say that the points AB lie on the line a or belong to the line a. And we can say that the line a passes through the points A and B.
The simplest geometric figures on a plane are a segment, a ray, a broken line.
A segment is a part of a line, which consists of all points of this line, bounded by two selected points. These points are the ends of the segment. A segment is indicated by indicating its ends.
A ray or half-line is a part of a line, which consists of all points of this line, lying on one side of its given point. This point is called the starting point of the half-line or the beginning of the ray. A ray has a start point but no end point.
Half-lines or rays are denoted by two lowercase Latin letters: the initial and any other letter corresponding to a point belonging to the half-line. In this case, the starting point is placed in the first place.
It turns out that the line is infinite: it has neither beginning nor end; a ray has only a beginning but no end, while a segment has a beginning and an end. Therefore, we can only measure a segment.
Several segments that are sequentially connected to each other so that the segments (neighboring) having one common point are not located on the same straight line represent a broken line.
The polyline can be closed or open. If the end of the last segment coincides with the beginning of the first, we have a closed broken line, if not, an open one.
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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
Although 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 a point, a line and a plane, they are given without a definition, but definitions of others geometric shapes 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, 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 future, 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 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 exposition 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.
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