What does a closed broken line look like? Point, line, straight line, ray, segment, polyline Looks like a polyline 3 links

A broken line in geometry is usually called a geometric figure, which consists of two or more segments. The end of one segment is the beginning of another. An obligatory condition that any broken line obeys is that adjacent segments should not be located on the same straight line.

These geometric figures are widely used in various fields of science and practice:

1. Cartography - for building images of streets and route maps.
2. Architecture - the outlines of buildings and structures.
3. Landscaping - decorative design and location of paths.
4. Chemistry - molecular structure of complex polymer compounds.
5. Medicine - monitors for monitoring the functional state of organs and systems.

Types of polylines

Considered geometric figures can be arranged in a variety of ways- they can be open and closed, intersecting and non-intersecting.

A closed polyline corresponds to a certain geometric figure - a polygon.

If the segments of one such figure have points of intersection with each other, this line is called self-intersecting.

In total, there are 4 types of similar lines in their structure:

1. Closed, which do not have intersections.
2. Open, which do not have intersections.
3. Non-closed self-intersecting.
4. Closed, having self-intersections.

A variation of such a geometric figure can be considered a zigzag, in which successive segments form a right angle and are parallel to each other through one. Zigzags are widely used in everyday life - in tailoring, decorative art, and the design of household items.

Features of closed lines

Let us consider in more detail the constituent parts of this geometric figure.

1. One segment of those that make up the described figure is called its link. A broken line can be considered such a line, which is made up of at least two segments - links. If there is one link, it is just a single segment.
2. There is also the concept of a polyline vertex. This term is usually called the point at which the ends of two links are connected. Such points in geometry are usually denoted using capital Latin letters. The polyline itself is called a combination of the notation of these vertices. For example, the combination ABCDEF can serve as the name of such a line.
3. If the ends of the extreme links of this geometric object are connected at one point, such a line is called closed.
4. The end vertices of such a figure in geometry are usually called black dots.

As mentioned above, this kind of lines can have self-intersections. The most popular example of a closed line that intersects itself is the five-pointed star.

Polygon as a kind of closed polyline

A variation of the described geometric figure is a polygon. The points in a polygon are its vertices, and the segments are called sides.

1. If the vertices belong to the same side of the polygon, they are called adjacent.
2. If a segment connects any two vertices that are not adjacent, it is called a diagonal.
3. If a polygon has n vertices, it is called an n-gon. Such a figure has the number of sides equal to n.
4. Such a broken line divides the plane into 2 parts - external and internal.
5. If the points of a polygon lie on one side of a straight line and pass through 2 adjacent vertices, it is commonly called convex.
6. The angle of a convex polygon at a given vertex is the angle that is formed by its two sides, for which this vertex is common.
7. The exterior angle of a convex polygon at a particular vertex is the angle adjacent to the interior angle of the polygon at that same vertex.

Examples of polygons are quadrangles, triangles, pentagons. Let us consider in more detail the distinctive features of these figures.

Triangle- This is a geometric figure, which consists of three points located not on the same straight line. These points are connected in pairs by segments.

quadrilateral In geometry, a figure is called a figure that has four corners and four sides. There are a wide variety of quadrilaterals - it can be trapezoids, squares, parallelograms, rhombuses.

At trapeze two sides are parallel and are called bases. The other two sides are not parallel. A parallelogram has two opposite sides parallel to each other.

A distinctive feature of a rectangle is that all its corners are right. A square has all four sides equal. In addition, all corners of a square are right angles.

If a polygon has all sides and angles equal, it is called regular. Such a polygon will always be convex.

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 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

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 beam begins, and the second is the point lying on the beam

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

A task: which broken line is longer, but 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.

Lesson duration: 35 minutes

Lesson type: Studying and primary consolidation of new material.

Target: Introduce the polyline and its components.

Lesson objectives:

1) Educational:

• introduce students to the broken line and its types; mastering the concepts of "broken line", "link of a broken line", "top of a broken line";
• repeat: segments, lines;
• improvement of computational skills and abilities.

2) Developing:

• develop logical thinking, spatial imagination, attention, memory, fantasy;
• improve the level of development of mathematical speech
• show the interdisciplinary connection between mathematics and astronomy.

3) Educators:

• educate the communication skills of students
• to cultivate pride in their homeland, achievements in science, technology, astronautics.

Materials and equipment:

1. multimedia presentation
2. Computer, projector, screen
3. "Study Route Sheet"
4. Pencils: yellow, blue, red
5. Spaghetti, a piece of plasticine
6. Massage mats for feet, SU-JOK (massage set "Chestnut" for hands)

Leading activity: productive, creative, problematic

Working methods: explanatory and illustrative, partially search, verbal, visual, practical.

Teacher function: organizer of cooperation; prospecting consultant.

Pedagogical technologies:

student-centered learning;

Explanatory and illustrative teaching;

Pedagogy of cooperation (learning dialogue);

ICT technology (presentation).

Expected Result:

• know what a broken line is, what it consists of, how it differs from a segment, a ray, a straight line, a curved line
• expanding knowledge of geometric material
• increasing the activity of students in the classroom
• use by students of acquired knowledge and skills in practical activities
• vocabulary enrichment

List of used literature.

1. Istomina N.B. Mathematics: a textbook for the 1st grade of educational institutions. - Smolensk: "Association XXI century", 2008.

2. Istomina N.B. Workbook for the textbook "Mathematics" for grade 1

During the classes

1. Organizing moment

Teacher: Children, 2011 has been declared the year of Russian cosmonautics in our country. Who among you is interested in space? Who wants to fly into space? Today is such an opportunity for the whole class. We will make a training flight. In order not to make mistakes during the flight, you need to prepare, restore some knowledge. What do you think we need to remember?

Children: Review numbers, addition and subtraction.

Teacher: I agree with you children. I will add: you need to know the geometric shapes passed.

2. Actualization of previous knowledge

Teacher: There are "Study Route Sheets" on your tables. All the results of the work in the lesson will be recorded on these sheets.

Get to know a new word. "Astronomy" (other Greek) is formed from the ancient Greek words "astron" - star and "nomos" - law or culture, and literally means "Law of the stars."

All scientists - astronomers know mathematics perfectly. Without this knowledge, it is impossible to accurately calculate the distances to distant stars, during the construction of spaceships, their trajectory of movement, development of speed:

So, the first task: "mathematical dictation". Listen to the condition, calculate in your mind, write down only the answer.

Of the 9 planets in the solar system, only two have female names. And how many male names are in the names of the planets of the solar system? (7)

The constellation Ursa Major has 7 bright stars. And in the constellation "Cassiopeia" 5 bright stars. How many more bright stars are there in the constellation Ursa Major? (2)

To my question at the beginning of the lesson: "Who dreams of flying into space?" 3 girls and 7 boys answered "yes". How many kids in our class want to fly into space? (10)

Children: write down the answers in their "Training Route Sheets", and one student - the "commander of the cosmonaut detachment" is instructed to write the answers on the board. Then all the children check, compare their results with the answers written on the board.

• What are the shapes called? (point, triangle, curved line, straight line, segment)
• What is the difference between a ray and a segment?
• What is the difference between a straight line and a ray?

Why is the second figure called a triangle? (has three vertices and three sides)

Can the sides of a triangle be called segments? Why? (the sides of the triangle are segments, because the lines that form them have borders)

Teacher: In the "Training Route Sheet" find the red dot and build a beam. What tool is needed? (Ruler)

Connect the two blue dots. What figure did you get? (Section)

Draw a straight line through the yellow dot. Can you do another one? What else? (Yes!)

It is true that an infinite number of straight lines can be drawn through a single point.

3. Physical education(The guys do the exercises, standing at their desks)

One, two!
The speed of light!
Three four!
We are flying!
To distant planets
We want to get there soon!
To drive ships
To fly into the sky
There is a lot to know.
You have to know a lot!
And at the same time, and at the same time
You notice,
Very important science
Maths!

4. Introduction of new material

Today we continue our journey to Geometry.

See what I have in my hands? (Vermicelli Spaghetti)

What geometric shape does it remind you of? (straight line)

Pick up the spaghetti handed out to you by the attendant. Break in the middle, and then break each part in half again.

What geometric shapes remind you of? (Segments, they turned out 4)

Connect them with pieces of plasticine to each other. Is it now possible to call the resulting figure a straight line? (Not)

What would you call such a geometric figure? (broken line)

I have to correct you a little, it's called a "broken" line.

See what a broken line consists of? (From segments)

Each broken line consists of several segments - links. How many links are in this broken line? (Four)

The links of the polyline do not lie on the same straight line. The end of one link is the beginning of another. The place where two links meet is called a vertex.

How many vertices does this polyline have? (Three)

In addition, a broken line has 2 ends.

5. Physical education- self-massage of the fingers with the help of a SU-JOK massager: Slide No. 4

In order
All planets
Call any of us:
Once - Mercury,
Two - Venus,
Three - Earth,
Four - Mars,
Five - Jupiter,
Six - Saturn.
Seven - Uranus,
The eighth is Neptune.
And after him, later
called Pluto.

6. Primary fastening

Teacher: Children, let's remember again, what are the curved lines? (closed and open)

What do you think, broken lines can be closed and open?

The teacher opens table number 1 on the board:

What figures are shown in the table? (broken lines)

Which broken line has the most links? (No. 4)

Which polyline has the fewest links? (No. 1)

Which polyline has three vertices? (No. 2)

Which polyline has five vertices? (No. 4)

The teacher opens table number 2 on the board:

Teacher: These are also broken lines. How do they differ from the broken lines in the first table? (All links are interconnected)

Such broken lines are called "closed" lines, and the lines on the first table are called "open" lines.

Name the closed broken line that has the fewest links. (#1)

That's right, but can there be a closed line of two links, think about it. Let's build such a broken line. (No, to "close" the line you need a third link)

Teacher: Find and name the constellations on the starry sky map: open broken lines and closed ones.

Teacher: If your "broken line of spaghetti" lying on the desk is turned over, it will resemble the constellation "Cassiopeia". It was named after the queen, who was bewitched by an insidious sorceress.

7. Physical education.

For eyes. Children follow the movement of Kolobok on Slide No. 4

Attention task

For a few seconds I will show you one figure. You must memorize it and lay out exactly the same from the counting sticks.

Now work in pairs. Check your classmate's attention.

What figure did you get?

What else can you say about her? Can it be called a broken line?

Can it be called closed? (open?) Why?

8. Summing up the lesson

What geometric figure are you familiar with? (broken line)

What elements does a broken line consist of? (From links and peaks)

What are broken lines? (Closed and open)

Turn over the Study Route Sheet. Circle with a colored pencil only broken lines, closed and open:

What line was described by Yuri Gagarin's ship in 108 minutes around the Earth? (open curved line)

In the lower right corner of the "Educational Route Sheet" an asterisk "smiles" at you. What geometric figure does it resemble? (Closed polyline) Determine the number of vertices? Links? Are there any ends?

Self-assessment of the work of students in the lesson:

You have 3 colored pencils. Color the star in green if you are completely satisfied with your work in the lesson; yellow - satisfied, but not completely; red - you have to try!

Additional material(Slides 18 - 31): information about planets, stars, space exploration.

1. How to measure the distance to the damage site with a REIS reflectometer

cable line, consisting of several cables of different types?

Any of the REIS reflectometers allows you to perform these measurements. In this case, two cases are possible.

1st case

with the same reduction factors.

In this case, the measurement of the distance to the damage site is carried out in the usual way. First, a shortening factor is set in the REIS reflectometer, which is the same for all pieces of the cable. Then one of the cursors is set to the beginning of the front of the probing pulse, and the other - to the beginning of the pulse reflected from the damage site. The distance between the cursors will correspond to the distance to the damage site.

An example of this case is shown in the figure.

The figure indicates:

L1 - length of the first piece of cable (shortening factor g 1),

L2 - length of the second piece of cable (shortening factor g 1),

L3 - distance from the beginning of the third piece of cable to the fault (shortening factor g 1),

L is the distance from the beginning of the cable to the point of damage,

A - signal reflected from the junction of the first and second pieces of cable,

B - signal reflected from the junction of the second and third pieces of cable,

C - signal reflected from the damage site.

The amplitude of the signals A and B depends on the ratio of the wave impedances W1, W2 and W3 of the individual pieces of the cable. If the wave impedances of adjacent cable pieces are equal, then the reflection from their junction has a minimum amplitude. And vice versa. In the above trace, the wave impedance W2 of the second piece of cable is less than the wave impedance W1 of the first piece of cable (W2< W1). Волновое сопротивление третьего и второго кусков кабеля также не равны, причем W3 >W2.

2nd case. The cable line consists of several pieces

with different reduction factors.

The measurement of the distance to the damage in this case is carried out in stages. Consider the sequence of measurements on the example of the reflectogram shown in the figure.

First, in the REIS reflectometer, the shortening factor g 1 is set for the first piece of cable and the length of this piece is measured. To do this, the zero cursor is set to the beginning of the front of the probing pulse (in Position 1), and the measuring cursor is set to the beginning of the front of the pulse reflected from the junction of the first and second pieces of the cable (in Position 2). The resulting length of the first piece of cable L1 is recorded.

Next, set the coefficient of shortening g 2 for the second piece of cable and measure the length of the second piece. To do this, leaving the measuring cursor in place, move the zero cursor to the beginning of the pulse reflected from the junction of the second and third pieces of the cable (to Position 3). The resulting length of the second piece of cable is recorded.

Then set the shortening factor g 3 for the third piece of cable and measure the distance from the beginning of the third piece of cable to the fault. To do this, leaving the zero cursor in place (in Position 3), move the measuring cursor to the beginning of the pulse reflected from the damage site (in Position 4). The resulting distance L3 from the beginning of the third piece of cable to the fault is recorded.

The distance to the damage point L is determined as the sum of the measured values: L = L1 + L2 + L3.

Similarly, you can determine the distance to the point of damage to the cable line, consisting of any number of pieces of cables of different types, with different coefficients of shortening.

2. Why is sometimes the length of the power cable on the reel specified by the manufacturer

cable differs from the length measured by the reflectometer? When measuring

the velocity factor has been set correctly. What is the length data

cables are more accurate?

Such a difference can be observed when the manufacturer measures the length of the cable using the bridge method according to the resistance of the cores. The cores in the power cable are twisted, so their length is always slightly longer than the length of the cable itself. Measuring the length of the cable by the resistance of the cores (electrical length) gives an overestimated value compared to the actual, geometric length of the cable.

The difference can also be in the case when the factory measures the length of the manufactured cable using mechanical devices that have rollers that can slip when the cable passes through them.

If the length of the power cable is measured with a reflectometer, then the discrepancy between the electrical and geometric lengths of the cable is taken into account in the shortening factor. Therefore, with a properly set velocity factor, length measurements made with a reflectometer are more accurate than measurements made with a bridge method.

Note: The above length discrepancy can be observed not only for the power cable, but also for any other cable.

3. Why, when measuring with a reflectometer over long (more than a few kilometers)

multi-pair telephone lines, such as the CCI type, zero line

trace curves and does not allow you to set

Is there a high gain in the reflectometer?

The indicated curvature of the zero line of the reflectogram, due to its characteristic appearance, is also called “skiing”. An example of such a “ski” is shown in the figure.

The figure shows the case in which in the “ski” area there is a signal reflected from the cable defect, in particular, a leak. When measuring with an OTDR on a cable, it is usually necessary to increase the gain due to the effect of attenuation. An increase in gain in the presence of a "ski" leads to further distortion of the reflectogram, which greatly complicates and may make analysis of the reflectogram completely impossible.

The reason for the appearance of the “ski” is the distributed capacitance of the cable (capacitance between the cores and between the core and the ground) and the longitudinal ohmic resistance of the cable cores.

At the moment of impact on the cable of the probing pulse from the reflectometer, the specified distributed capacitance of the cable is charged. At the end of the probing pulse, the distributed capacitance of the cable begins to gradually discharge, a “ski” appears.

To reduce the influence of the "ski" on the results of measurements with REIS-105, REIS-205 or REIS-305 reflectometers, you need to turn on the compensation pulse and select its duration.

The degree of compensation can be set by the operator depending on the line, since the “ski” depends on many cable parameters: the number and diameter of the cores, the length of the cable, the type of insulation, etc.

4. When measuring the length of an armored cable with a reflectometer, we get

the following incomprehensible results: if you connect the reflectometer according to the scheme

core-core, then the cable length is less than when connecting

according to the scheme lived-armor. What's the matter here?

In fact, no matter how you connect the reflectometer to the cable when measuring its length, the length of the cable remains the same.

The different values ​​of the cable lengths you measured for different connection schemes are due to the fact that the coefficients of shortening of the wave channels core-core and core-armor differ from each other.

In this lesson, we will get acquainted with the concepts of "closed line" and "open line", learn how to distinguish and build them. We will also consider such concepts as “links” and “vertices” of a curved line. In the future, we will use this knowledge to solve more complex problems.

Topic:Getting to know the basic concepts

Lesson: Closed and Open Lines

Exercise 1

In this figure, we see that it will be easier for the sheep to get out of the first fence, because it is open - not closed. The second fence will make it harder to get out, as it is closed. Let's draw lines that will correspond to the first and second fences.

So, we got two lines, of which the first is closed, and the second is not closed.

Task 2: Determine which lines in Fig. 3 are closed and which are not closed.

In the figure we see that lines No. 1, 3, 6 are open lines. In order to close these lines, it is enough to connect the ends of the lines together. We get:

So, a line whose ends are not joined together is called open line. A line whose ends are joined together is called closed line.

Each broken line consists of several segments - links . The links of the polyline do not lie on the same straight line. The end of one link is the beginning of another. The place where two links are connected, as well as the ends of an open broken line, is called summit .

So, in this lesson, we got acquainted with the concepts of "closed line" and "open line". We have learned how to build them, as well as apply knowledge in practice to build such lines.

Bibliography

1. Aleksandrova L.A., Mordkovich A.G. Mathematics 1st grade. - M: Mnemosyne, 2012.
2. Bashmakov M.I., Nefedova M.G. Maths. 1 class. - M: Astrel, 2012.
3. Bedenko M.V. Maths. 1 class. - M7: Russian word, 2012.

1. Festival of pedagogical ideas ().

3. Festival of pedagogical ideas ().

Homework

1. Determine which lines are shown in the figure.

2. Determine the number of links of each line.

3. Determine the number of vertices for each line.

4. Draw an open line with 4 vertices.

5. Build a closed line with 6 links.