Lesson "Polygons. Types of polygons" within the technology "Development of critical thinking through reading and writing"
Polygon Properties
A polygon is a geometric figure, usually defined as a closed polyline without self-intersections (a simple polygon (Fig. 1a)), but sometimes self-intersections are allowed (then the polygon is not simple).
The vertices of the polyline are called the vertices of the polygon, and the segments are called the sides of the polygon. The vertices of a polygon are called neighbors if they are the ends of one of its sides. Line segments connecting non-neighboring vertices of a polygon are called diagonals.
An angle (or internal angle) of a convex polygon at a given vertex is the angle formed by its sides converging at this vertex, and the angle is considered from the side of the polygon. In particular, the angle may exceed 180° if the polygon is not convex.
The exterior angle of a convex polygon at a given vertex is the angle adjacent to the interior angle of the polygon at that vertex. In general, the outside angle is the difference between 180° and the inside angle. From each vertex of the -gon for > 3, there are - 3 diagonals, so the total number of diagonals of the -gon is equal.
A polygon with three vertices is called a triangle, with four - a quadrilateral, with five - a pentagon, and so on.
Polygon with n peaks is called n- square.
A flat polygon is a figure that consists of a polygon and the finite part of the area bounded by it.
A polygon is called convex if one of the following (equivalent) conditions is met:
- 1. it lies on one side of any straight line connecting its neighboring vertices. (i.e., the extensions of the sides of a polygon do not intersect its other sides);
- 2. it is the intersection (i.e. common part) of several half-planes;
- 3. any segment with ends at points belonging to the polygon belongs entirely to it.
A convex polygon is called regular if all sides are equal and all angles are equal, for example, an equilateral triangle, a square, and a pentagon.
A convex polygon is said to be inscribed about a circle if all its sides are tangent to some circle
A regular polygon is a polygon in which all angles and all sides are equal.
Polygon properties:
1 Each diagonal of a convex -gon, where >3, decomposes it into two convex polygons.
2 The sum of all angles of a convex -gon is equal to.
D-in: Let's prove the theorem by the method of mathematical induction. For = 3 it is obvious. Assume that the theorem is true for a -gon, where <, and prove it for -gon.
Let be a given polygon. Draw a diagonal of this polygon. By Theorem 3, the polygon is decomposed into a triangle and a convex -gon (Fig. 5). By the induction hypothesis. On the other hand, . Adding these equalities and taking into account that (- inner beam angle ) and (- inner beam angle ), we get. When we get: .
3 About any regular polygon it is possible to describe a circle, and moreover, only one.
D-in: Let a regular polygon, and and be the bisectors of the angles, and (Fig. 150). Since, therefore, * 180°< 180°. Отсюда следует, что биссектрисы и углов и пересекаются в некоторой точке O. Let's prove that O = OA 2 = O =… = OA P . Triangle O isosceles, therefore O= O. According to the second criterion for the equality of triangles, therefore, O = O. Similarly, it is proved that O = O etc. So the point O equidistant from all vertices of the polygon, so the circle with the center O radius O is circumscribed about a polygon.
Let us now prove that there is only one circumscribed circle. Consider some three vertices of a polygon, for example, BUT 2 , . Since only one circle passes through these points, then about the polygon … You cannot describe more than one circle.
- 4 In any regular polygon, you can inscribe a circle and, moreover, only one.
- 5 A circle inscribed in a regular polygon touches the sides of the polygon at their midpoints.
- 6 The center of a circle circumscribing a regular polygon coincides with the center of a circle inscribed in the same polygon.
- 7 Symmetry:
A figure is said to be symmetric (symmetrical) if there is such a movement (not identical) that transforms this figure into itself.
- 7.1. A general triangle has no axes or centers of symmetry, it is not symmetrical. An isosceles (but not equilateral) triangle has one axis of symmetry: the perpendicular bisector to the base.
- 7.2. An equilateral triangle has three axes of symmetry (perpendicular bisectors to the sides) and rotational symmetry about the center with a rotation angle of 120°.
7.3 Any regular n-gon has n axes of symmetry, all of which pass through its center. It also has rotational symmetry about the center with a rotation angle.
Even n some axes of symmetry pass through opposite vertices, others through the midpoints of opposite sides.
For odd n each axis passes through the vertex and midpoint of the opposite side.
The center of a regular polygon with an even number of sides is its center of symmetry. A regular polygon with an odd number of sides has no center of symmetry.
8 Similarity:
With similarity, and -gon goes into a -gon, half-plane - into a half-plane, therefore convex n-gon becomes convex n-gon.
Theorem: If the sides and angles of convex polygons and satisfy the equalities:
where is the podium coefficient
then these polygons are similar.
- 8.1 The ratio of the perimeters of two similar polygons is equal to the coefficient of similarity.
- 8.2. The ratio of the areas of two convex similar polygons is equal to the square of the similarity coefficient.
polygon triangle perimeter theorem
Sections: Maths
Subject, age of students: geometry, grade 9
The purpose of the lesson: the study of types of polygons.
Learning task: to update, expand and generalize students' knowledge of polygons; form an idea of the “components” of a polygon; conduct a study of the number of constituent elements of regular polygons (from a triangle to n-gon);
Developing task: to develop the ability to analyze, compare, draw conclusions, develop computational skills, oral and written mathematical speech, memory, as well as independence in thinking and learning activities, the ability to work in pairs and groups; develop research and educational activities;
Educational task: to cultivate independence, activity, responsibility for the task assigned, perseverance in achieving the goal.
During the classes: a quote is written on the blackboard
“Nature speaks the language of mathematics, the letters of this language ... mathematical figures.” G. Gallilei
At the beginning of the lesson, the class is divided into working groups (in our case, the division into groups of 4 people each - the number of group members is equal to the number of question groups).
1. Call stage-
Goals:
a) updating students' knowledge on the topic;
b) the awakening of interest in the topic under study, the motivation of each student for learning activities.
Reception: The game "Do you believe that ...", organization of work with text.
Forms of work: frontal, group.
“Do you believe that….”
1. ... the word "polygon" indicates that all the figures of this family have "many corners"?
2. … does a triangle belong to a large family of polygons that stand out among many different geometric shapes on a plane?
3. …is a square a regular octagon (four sides + four corners)?
Today in the lesson we will talk about polygons. We learn that this figure is bounded by a closed broken line, which in turn can be simple, closed. Let's talk about the fact that polygons are flat, regular, convex. One of the flat polygons is a triangle that you have been familiar with for a long time (you can show students posters depicting polygons, a broken line, show their various types, you can also use TCO).
2. Stage of comprehension
Purpose: obtaining new information, its comprehension, selection.
Reception: zigzag.
Forms of work: individual->pair->group.
Each group is given a text on the topic of the lesson, and the text is designed in such a way that it includes both information already known to students and completely new information. Together with the text, students receive questions, the answers to which must be found in this text.
Polygons. Types of polygons.
Who hasn't heard of the mysterious Bermuda Triangle, where ships and planes disappear without a trace? But the triangle familiar to us from childhood is fraught with a lot of interesting and mysterious things.
In addition to the types of triangles already known to us, divided by sides (scalene, isosceles, equilateral) and angles (acute-angled, obtuse-angled, right-angled), the triangle belongs to a large family of polygons distinguished from many different geometric shapes on the plane.
The word "polygon" indicates that all the figures of this family have "many corners". But this is not enough to characterize the figure.
A broken line A 1 A 2 ... A n is a figure that consists of points A 1, A 2, ... A n and segments A 1 A 2, A 2 A 3, ... connecting them. The points are called the vertices of the polyline, and the segments are called the links of the polyline. (fig.1)
A broken line is called simple if it does not have self-intersections (Fig. 2,3).
A broken line is called closed if its ends coincide. The length of a broken line is the sum of the lengths of its links (Fig. 4).
A simple closed broken line is called a polygon if its adjacent links do not lie on the same straight line (Fig. 5).
Substitute in the word “polygon” instead of the “many” part a specific number, for example 3. You will get a triangle. Or 5. Then - a pentagon. Note that there are as many angles as there are sides, so these figures could well be called multilaterals.
The vertices of the polyline are called the vertices of the polygon, and the links of the polyline are called the sides of the polygon.
The polygon divides the plane into two regions: internal and external (Fig. 6).
A plane polygon or polygonal region is a finite part of a plane bounded by a polygon.
Two vertices of a polygon that are ends of the same side are called neighbors. Vertices that are not ends of one side are non-adjacent.
A polygon with n vertices and therefore n sides is called an n-gon.
Although the smallest number of sides of a polygon is 3. But triangles, connecting with each other, can form other shapes, which in turn are also polygons.
Segments connecting non-neighboring vertices of a polygon are called diagonals.
A polygon is called convex if it lies in one half-plane with respect to any line containing its side. In this case, the straight line itself is considered to belong to the half-plane.
The angle of a convex polygon at a given vertex is the angle formed by its sides converging at that vertex.
Let's prove the theorem (on the sum of angles of a convex n-gon): The sum of the angles of a convex n-gon is equal to 180 0 *(n - 2).
Proof. In the case n=3 the theorem is true. Let А 1 А 2 …А n be a given convex polygon and n>3. Let's draw diagonals in it (from one vertex). Since the polygon is convex, these diagonals divide it into n - 2 triangles. The sum of the angles of the polygon is the same as the sum of the angles of all these triangles. The sum of the angles of each triangle is 180 0, and the number of these triangles is n - 2. Therefore, the sum of the angles of a convex n - angle A 1 A 2 ... A n is 180 0 * (n - 2). The theorem has been proven.
The exterior angle of a convex polygon at a given vertex is the angle adjacent to the interior angle of the polygon at that vertex.
A convex polygon is called regular if all sides are equal and all angles are equal.
So the square can be called differently - a regular quadrilateral. Equilateral triangles are also regular. Such figures have long been of interest to the masters who decorated the buildings. They made beautiful patterns, for example, on the parquet. But not all regular polygons could be used to form parquet. Parquet cannot be formed from regular octagons. The fact is that they have each angle equal to 135 0. And if any point is the vertex of two such octagons, then they will have 270 0, and there is nowhere for the third octagon to fit: 360 0 - 270 0 \u003d 90 0. But enough for a square. Therefore, it is possible to fold the parquet from regular octagons and squares.
The stars are correct. Our five-pointed star is a regular pentagonal star. And if you rotate the square around the center by 45 0, you get a regular octagonal star.
1 group
What is a broken line? Explain what vertices and links of a polyline are.
Which broken line is called simple?
Which broken line is called closed?
What is a polygon? What are the vertices of a polygon called? What are the sides of a polygon?
2 group
What is a flat polygon? Give examples of polygons.
What is n-gon?
Explain which vertices of the polygon are adjacent and which are not.
What is the diagonal of a polygon?
3 group
What is a convex polygon?
Explain which corners of the polygon are external and which are internal?
What is a regular polygon? Give examples of regular polygons.
4 group
What is the sum of the angles of a convex n-gon? Prove it.
Students work with the text, look for answers to the questions posed, after which expert groups are formed, in which work is carried out on the same issues: students highlight the main thing, draw up a supporting abstract, present information in one of the graphic forms. At the end of the work, students return to their working groups.
3. Stage of reflection -
a) assessment of their knowledge, challenge to the next step of knowledge;
b) understanding and appropriation of the received information.
Reception: research work.
Forms of work: individual->pair->group.
The working groups are experts in the answers to each of the sections of the proposed questions.
Returning to the working group, the expert introduces the other members of the group with the answers to their questions. In the group there is an exchange of information of all members of the working group. Thus, in each working group, thanks to the work of experts, a general idea is formed on the topic under study.
Research work of students - filling in the table.
Regular polygons | Drawing | Number of sides | Number of peaks | Sum of all internal angles | Degree measure int. angle | Degree measure of external angle | Number of diagonals |
A) a triangle | |||||||
B) quadrilateral | |||||||
B) five-wall | |||||||
D) hexagon | |||||||
E) n-gon |
Solving interesting problems on the topic of the lesson.
- In the quadrilateral, draw a line so that it divides it into three triangles.
- How many sides does a regular polygon have, each of whose interior angles is equal to 135 0 ?
- In a certain polygon, all interior angles are equal to each other. Can the sum of the interior angles of this polygon be: 360 0 , 380 0 ?
Summing up the lesson. Recording homework.
Triangle, square, hexagon - these figures are known to almost everyone. But not everyone knows what a regular polygon is. But this is all the same Regular polygon is called the one that has equal angles and sides. There are a lot of such figures, but they all have the same properties, and the same formulas apply to them.
Properties of regular polygons
Any regular polygon, be it a square or an octagon, can be inscribed in a circle. This basic property is often used when constructing a figure. In addition, a circle can also be inscribed in a polygon. In this case, the number of points of contact will be equal to the number of its sides. It is important that a circle inscribed in a regular polygon will have a common center with it. These geometric figures are subject to the same theorems. Any side of a regular n-gon is associated with the radius R of the circumscribed circle around it. Therefore, it can be calculated using the following formula: a = 2R ∙ sin180°. Through you can find not only the sides, but also the perimeter of the polygon.
How to find the number of sides of a regular polygon
Any one consists of a certain number of segments equal to each other, which, when connected, form a closed line. In this case, all the corners of the formed figure have the same value. Polygons are divided into simple and complex. The first group includes a triangle and a square. Complex polygons have more sides. They also include star-shaped figures. For complex regular polygons, the sides are found by inscribing them in a circle. Let's give a proof. Draw a regular polygon with an arbitrary number of sides n. Describe a circle around it. Specify the radius R. Now imagine that some n-gon is given. If the points of its angles lie on a circle and are equal to each other, then the sides can be found by the formula: a = 2R ∙ sinα: 2.
Finding the number of sides of an inscribed right triangle
An equilateral triangle is a regular polygon. The same formulas apply to it as to the square and the n-gon. A triangle will be considered correct if it has the same length sides. In this case, the angles are 60⁰. Construct a triangle with given side length a. Knowing its median and height, you can find the value of its sides. To do this, we will use the method of finding through the formula a \u003d x: cosα, where x is the median or height. Since all sides of the triangle are equal, we get a = b = c. Then the following statement is true: a = b = c = x: cosα. Similarly, you can find the value of the sides in an isosceles triangle, but x will be the given height. At the same time, it should be projected strictly on the base of the figure. So, knowing the height x, we find the side a of an isosceles triangle using the formula a \u003d b \u003d x: cosα. After finding the value of a, you can calculate the length of the base c. Let's apply the Pythagorean theorem. We will look for the value of half the base c: 2=√(x: cosα)^2 - (x^2) = √x^2 (1 - cos^2α) : cos^2α = x ∙ tgα. Then c = 2xtanα. In such a simple way, you can find the number of sides of any inscribed polygon.
Calculating the sides of a square inscribed in a circle
Like any other inscribed regular polygon, a square has equal sides and angles. The same formulas apply to it as to the triangle. You can calculate the sides of a square using the value of the diagonal. Let's consider this method in more detail. It is known that the diagonal bisects the angle. Initially, its value was 90 degrees. Thus, after division, two are formed. Their angles at the base will be equal to 45 degrees. Accordingly, each side of the square will be equal, that is: a \u003d b \u003d c \u003d d \u003d e ∙ cosα \u003d e √ 2: 2, where e is the diagonal of the square, or the base of the right triangle formed after division. This is not the only way to find the sides of a square. Let's inscribe this figure in a circle. Knowing the radius of this circle R, we find the side of the square. We will calculate it as follows a4 = R√2. The radii of regular polygons are calculated by the formula R \u003d a: 2tg (360 o: 2n), where a is the length of the side.
How to calculate the perimeter of an n-gon
The perimeter of an n-gon is the sum of all its sides. It is easy to calculate it. To do this, you need to know the values of all sides. For some types of polygons, there are special formulas. They allow you to find the perimeter much faster. It is known that any regular polygon has equal sides. Therefore, in order to calculate its perimeter, it is enough to know at least one of them. The formula will depend on the number of sides of the figure. In general, it looks like this: P \u003d an, where a is the value of the side, and n is the number of angles. For example, to find the perimeter of a regular octagon with a side of 3 cm, you need to multiply it by 8, that is, P = 3 ∙ 8 = 24 cm. For a hexagon with a side of 5 cm, we calculate as follows: P = 5 ∙ 6 = 30 cm. And so for each polygon.
Finding the perimeter of a parallelogram, square and rhombus
Depending on how many sides a regular polygon has, its perimeter is calculated. This makes the task much easier. Indeed, unlike other figures, in this case it is not necessary to look for all its sides, just one is enough. By the same principle, we find the perimeter of quadrangles, that is, a square and a rhombus. Despite the fact that these are different figures, the formula for them is the same P = 4a, where a is the side. Let's take an example. If the side of a rhombus or square is 6 cm, then we find the perimeter as follows: P \u003d 4 ∙ 6 \u003d 24 cm. A parallelogram has only opposite sides. Therefore, its perimeter is found using a different method. So, we need to know the length a and the width b of the figure. Then we apply the formula P \u003d (a + c) ∙ 2. A parallelogram, in which all sides and angles between them are equal, is called a rhombus.
Finding the perimeter of an equilateral and right triangle
The perimeter of the correct one can be found by the formula P \u003d 3a, where a is the length of the side. If it is unknown, it can be found through the median. In a right triangle, only two sides are equal. The basis can be found through the Pythagorean theorem. After the values of all three sides become known, we calculate the perimeter. It can be found by applying the formula P \u003d a + b + c, where a and b are equal sides, and c is the base. Recall that in an isosceles triangle a \u003d b \u003d a, therefore, a + b \u003d 2a, then P \u003d 2a + c. For example, the side of an isosceles triangle is 4 cm, find its base and perimeter. We calculate the value of the hypotenuse according to the Pythagorean theorem c \u003d √a 2 + in 2 \u003d √16 + 16 \u003d √32 \u003d 5.65 cm. Now we calculate the perimeter P \u003d 2 ∙ 4 + 5.65 \u003d 13.65 cm.
How to find the angles of a regular polygon
A regular polygon occurs in our lives every day, for example, an ordinary square, triangle, octagon. It would seem that there is nothing easier than building this figure yourself. But this is just at first glance. In order to construct any n-gon, you need to know the value of its angles. But how do you find them? Even scientists of antiquity tried to build regular polygons. They guessed to fit them into circles. And then the necessary points were marked on it, connected by straight lines. For simple figures, the construction problem has been solved. Formulas and theorems have been obtained. For example, Euclid in his famous work "The Beginning" was engaged in solving problems for 3-, 4-, 5-, 6- and 15-gons. He found ways to construct them and find angles. Let's see how to do this for a 15-gon. First you need to calculate the sum of its internal angles. It is necessary to use the formula S = 180⁰(n-2). So, we are given a 15-gon, which means that the number n is 15. We substitute the data we know into the formula and get S = 180⁰ (15 - 2) = 180⁰ x 13 = 2340⁰. We have found the sum of all interior angles of a 15-gon. Now we need to get the value of each of them. There are 15 angles in total. We do the calculation of 2340⁰: 15 = 156⁰. This means that each internal angle is 156⁰, now using a ruler and a compass, you can build a regular 15-gon. But what about more complex n-gons? For centuries, scientists have struggled to solve this problem. It was only found in the 18th century by Carl Friedrich Gauss. He was able to build a 65537-gon. Since then, the problem has officially been considered completely solved.
Calculation of angles of n-gons in radians
Of course, there are several ways to find the corners of polygons. Most often they are calculated in degrees. But you can also express them in radians. How to do it? It is necessary to proceed as follows. First, we find out the number of sides of a regular polygon, then subtract 2 from it. So, we get the value: n - 2. Multiply the difference found by the number n (“pi” \u003d 3.14). Now it remains only to divide the resulting product by the number of angles in the n-gon. Consider these calculations using the example of the same fifteen-sided. So, the number n is 15. Let's apply the formula S = p(n - 2) : n = 3.14(15 - 2) : 15 = 3.14 ∙ 13: 15 = 2.72. This is of course not the only way to calculate an angle in radians. You can simply divide the size of the angle in degrees by the number 57.3. After all, that many degrees is equivalent to one radian.
Calculation of the value of angles in degrees
In addition to degrees and radians, you can try to find the value of the angles of a regular polygon in grads. This is done in the following way. Subtract 2 from the total number of angles, divide the resulting difference by the number of sides of a regular polygon. We multiply the result found by 200. By the way, such a unit of measurement of angles as degrees is practically not used.
Calculation of external corners of n-gons
For any regular polygon, in addition to the internal one, you can also calculate the external angle. Its value is found in the same way as for other figures. So, to find the outer corner of a regular polygon, you need to know the value of the inner one. Further, we know that the sum of these two angles is always 180 degrees. Therefore, we do the calculations as follows: 180⁰ minus the value of the internal angle. We find the difference. It will be equal to the value of the angle adjacent to it. For example, the inner corner of a square is 90 degrees, so the outer angle will be 180⁰ - 90⁰ = 90⁰. As we can see, it is not difficult to find it. The external angle can take a value from +180⁰ to, respectively, -180⁰.
A polygon is a geometric figure that is bounded on all sides by a closed broken line. In this case, the number of links of the polyline should not be less than three. Each pair of polyline segments has a common point and forms angles. The number of corners, together with the number of polyline segments, are the main characteristics of a polygon. In each polygon, the number of links of the bounding closed polyline is the same as the number of corners.
Sides in geometry are usually called links of a polyline that limits a geometric object. Vertices are the points of contact between two adjacent sides., by the number of which polygons get their names.
If a closed broken line consists of three segments, it is called a triangle; respectively, from four segments - a quadrangle, from five - a pentagon, etc.
To designate a triangle or quadrilateral, use capital Latin letters denoting its vertices. Letters are called in order - clockwise or counterclockwise.
Basic concepts
When describing the definition of a polygon, some related geometric concepts should be taken into account:
- If the vertices are ends of the same side, they are called neighbors.
- If a segment connects non-neighboring vertices, then it is called a diagonal. A triangle cannot have diagonals.
- An internal angle is an angle at one of the vertices, which is formed by its two sides converging at this point. It is always located in the inner region of the geometric figure. If the polygon is non-convex, its size can exceed 180 degrees.
- The external angle at a certain vertex is the angle adjacent to the internal one at it. In other words, the outer angle can be considered the difference between 180° and the value of the inner angle.
- The sum of the values of all segments is called the perimeter.
- If all sides and all angles are equal, it is called correct. Only convex ones can be correct.
As mentioned above, the names of polygonal geometries are based on the number of vertices. If a figure has n of them, it is called n-gon:
- A polygon is called flat if it limits the finite part of the plane. This geometric figure can be inscribed in a circle or circumscribed around a circle.
- An n-gon is called convex if it meets one of the conditions below.
- The figure is located on one side of a straight line that connects two adjacent vertices.
- This figure serves as a common part or intersection of several half-planes.
- The diagonals are located inside the polygon.
- If the ends of the segment are located at points that belong to the polygon, the entire segment belongs to it.
- A figure can be called regular if all its segments and all angles are equal. Examples are a square, an equilateral triangle, or a regular pentagon.
- If an n-gon is non-convex, all its sides and angles are equal, and the vertices coincide with those of a regular n-gon, it is called star-shaped. Such figures may have self-intersections. An example would be a pentagram or a hexagram.
- A triangle or quadrangle is said to be inscribed in a circle when all its vertices are located inside the same circle. If the sides of this figure have points of contact with the circle, this is a polygon circumscribed about some circle.
Any a convex n-gon can be divided into triangles. In this case, the number of triangles is less than the number of sides by 2.
Types of figures
It is a polygon with three vertices and three line segments connecting them. In this case, the connection points of the segments do not lie on one straight line.
The connection points of the segments are triangle vertices. The segments themselves are called the sides of the triangle. The total sum of the interior angles of each triangle is 180°.
According to the ratios between the sides, all triangles can be divided into several types:
- equilateral- in which the length of all segments is the same.
- Isosceles Triangles that have two out of three equal segments.
- Versatile- if the length of all segments is different.
In addition, it is customary to distinguish the following triangles:
- Acute-angled.
- Rectangular.
- obtuse.
quadrilateral
A quadrilateral is a flat figure that has 4 vertices and 4 segments that connect them in series.
- If all corners of a quadrilateral are right angles, the figure is called a rectangle.
- A rectangle in which all sides are the same size is called a square.
- A quadrilateral with all sides equal is called a rhombus.
Three vertices of a quadrilateral cannot lie on the same straight line.
Video
You can find more information about polygons in this video.
In this lesson, we will start a new topic and introduce a new concept for us - a "polygon". We will look at the basic concepts associated with polygons: sides, vertices, corners, convexity and non-convexity. Then we will prove the most important facts, such as the theorem on the sum of the interior angles of a polygon, the theorem on the sum of the exterior angles of a polygon. As a result, we will come close to studying special cases of polygons, which will be considered in future lessons.
Theme: Quadrangles
Lesson: Polygons
In the course of geometry, we study the properties of geometric shapes and have already considered the simplest of them: triangles and circles. At the same time, we also discussed specific special cases of these figures, such as right-angled, isosceles and regular triangles. Now it's time to talk about more general and complex shapes - polygons.
With a special case polygons we are already familiar - this is a triangle (see Fig. 1).
Rice. 1. Triangle
The name itself already emphasizes that this is a figure that has three corners. Therefore, in polygon there can be many of them, i.e. more than three. For example, let's draw a pentagon (see Fig. 2), i.e. figure with five corners.
Rice. 2. Pentagon. Convex polygon
Definition.Polygon- a figure consisting of several points (more than two) and the corresponding number of segments that connect them in series. These points are called peaks polygon, and segments - parties. In this case, no two adjacent sides lie on the same straight line and no two non-adjacent sides intersect.
Definition.regular polygon is a convex polygon in which all sides and angles are equal.
Any polygon divides the plane into two regions: internal and external. The interior is also referred to as polygon.
In other words, for example, when they talk about a pentagon, they mean both its entire inner region and its border. And the inner area also includes all points that lie inside the polygon, i.e. the point also belongs to the pentagon (see Fig. 2).
Polygons are sometimes also called n-gons to emphasize that the general case of having some unknown number of corners (n pieces) is being considered.
Definition. Polygon Perimeter is the sum of the lengths of the sides of the polygon.
Now we need to get acquainted with the types of polygons. They are divided into convex and non-convex. For example, the polygon shown in Fig. 2 is convex, and in Fig. 3 non-convex.
Rice. 3. Non-convex polygon
Definition 1. Polygon called convex, if when drawing a straight line through any of its sides, the entire polygon lies only on one side of this line. non-convex are all the rest polygons.
It is easy to imagine that when extending any side of the pentagon in Fig. 2 it will all be on one side of this straight line, i.e. he is convex. But when drawing a straight line through the quadrilateral in Fig. 3 we already see that it divides it into two parts, i.e. he is non-convex.
But there is another definition of the convexity of a polygon.
Definition 2. Polygon called convex if, when choosing any two of its interior points and connecting them with a segment, all points of the segment are also interior points of the polygon.
A demonstration of the use of this definition can be seen in the example of constructing segments in Fig. 2 and 3.
Definition. Diagonal A polygon is any segment that connects two non-adjacent vertices.
To describe the properties of polygons, there are two most important theorems about their angles: convex polygon interior angle sum theorem and convex polygon exterior angle sum theorem. Let's consider them.
Theorem. On the sum of interior angles of a convex polygon (n-gon).
Where is the number of its angles (sides).
Proof 1. Let's depict in Fig. 4 convex n-gon.
Rice. 4. Convex n-gon
Draw all possible diagonals from the vertex. They divide the n-gon into triangles, because each of the sides of the polygon forms a triangle, except for the sides adjacent to the vertex. It is easy to see from the figure that the sum of the angles of all these triangles will just be equal to the sum of the interior angles of the n-gon. Since the sum of the angles of any triangle is , then the sum of the interior angles of an n-gon is:
Q.E.D.
Proof 2. Another proof of this theorem is also possible. Let's draw a similar n-gon in Fig. 5 and connect any of its interior points to all vertices.
Rice. 5.
We got a partition of an n-gon into n triangles (how many sides, so many triangles). The sum of all their angles is equal to the sum of the interior angles of the polygon and the sum of the angles at the interior point, and this is the angle. We have:
Q.E.D.
Proven.
According to the proved theorem, it can be seen that the sum of the angles of an n-gon depends on the number of its sides (on n). For example, in a triangle, and the sum of the angles is . In a quadrilateral, and the sum of the angles - etc.
Theorem. On the sum of exterior angles of a convex polygon (n-gon).
Where is the number of its corners (sides), and , ..., are external corners.
Proof. Let's draw a convex n-gon in Fig. 6 and denote its internal and external angles.
Rice. 6. Convex n-gon with marked exterior corners
Because the outer corner is connected to the inner one as adjacent, then and similarly for other external corners. Then:
During the transformations, we used the already proven theorem on the sum of the interior angles of an n-gon.
Proven.
From the proved theorem follows an interesting fact that the sum of the external angles of a convex n-gon is equal to on the number of its angles (sides). By the way, unlike the sum of interior angles.
Bibliography
- Aleksandrov A.D. etc. Geometry, grade 8. - M.: Education, 2006.
- Butuzov V.F., Kadomtsev S.B., Prasolov V.V. Geometry, 8th grade. - M.: Education, 2011.
- Merzlyak A.G., Polonsky V.B., Yakir S.M. Geometry, 8th grade. - M.: VENTANA-GRAF, 2009.
- Profmeter.com.ua ().
- Narod.ru ().
- Xvatit.com().
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