A polygon with vertices is called a polygon. Lesson "Polygons. Types of polygons" within the framework of the technology "Development of critical thinking through reading and writing"

Types of polygons:

Quadrilaterals

Quadrilaterals, respectively, consist of 4 sides and angles.

Sides and angles opposite each other are called opposite.

Diagonals divide convex quadrilaterals into triangles (see picture).

The sum of the angles of a convex quadrilateral is 360° (using the formula: (4-2)*180°).

Parallelograms

Parallelogram is a convex quadrilateral with opposite parallel sides (number 1 in the figure).

Opposite sides and angles in a parallelogram are always equal.

And the diagonals at the intersection point are divided in half.

Trapeze

Trapezoid- this is also a quadrilateral, and in trapezoids Only two sides are parallel, which are called reasons. Other sides are sides.

The trapezoid in the figure is numbered 2 and 7.

As in a triangle:

If the sides are equal, then the trapezoid is isosceles;

If one of the angles is right, then the trapezoid is rectangular.

The midline of the trapezoid is equal to half the sum of the bases and is parallel to them.

Rhombus

Rhombus is a parallelogram in which all sides are equal.

In addition to the properties of a parallelogram, rhombuses have their own special property - The diagonals of a rhombus are perpendicular each other and bisect the corners of a rhombus.

In the picture there is a rhombus number 5.

Rectangles

Rectangle is a parallelogram in which each angle is right (see figure number 8).

In addition to the properties of a parallelogram, rectangles have their own special property - the diagonals of the rectangle are equal.

Squares

Square is a rectangle with all sides equal (No. 4).

It has the properties of a rectangle and a rhombus (since all sides are equal).

Properties of Polygons

A polygon is a geometric figure, usually defined as a closed broken line without self-intersections (a simple polygon (Fig. 1a)), but sometimes self-intersections are allowed (then the polygon is not simple).

The vertices of the polygon are called the vertices of the polygon, and the segments are called the sides of the polygon. The vertices of a polygon are called adjacent if they are the ends of one of its sides. The segments connecting non-adjacent vertices of a polygon are called diagonals.

The angle (or interior angle) of a convex polygon at a given vertex is the angle formed by its sides converging at this vertex, and the angle is calculated from the side of the polygon. In particular, the angle can exceed 180° if the polygon is non-convex.

The exterior angle of a convex polygon at a given vertex is the angle adjacent to the interior angle of the polygon at this vertex. In general, an exterior angle is the difference between 180° and an interior 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, etc.

Polygon with n called vertices n- square.

A flat polygon is a figure that consists of a polygon and a finite part of the area limited 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 the polygon do not intersect its other sides);
• 2. it is the intersection (i.e. the 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, square and pentagon.

A convex polygon is said to be circumscribed about a circle if all its sides touch some circle

A regular polygon is a polygon in which all angles and all sides are equal.

Properties of polygons:

1 Each diagonal of a convex -gon, where >3, decomposes it into two convex polygons.

2 The sum of all angles of a convex triangle is equal.

D-vo: We will prove the theorem using the method of mathematical induction. At = 3 it is obvious. Let us assume that the theorem is true for a -gon, where <, and prove it for -gon.

Let be a given polygon. Let's draw the diagonal of this polygon. According to Theorem 3, the polygon is decomposed into a triangle and a convex triangle (Fig. 5). By the induction hypothesis. On the other side, . Adding these equalities and taking into account that (- internal angle beam ) And (- internal angle beam ), we get. When we get: .

3 Around any regular polygon you can describe a circle, and only one.

D-vo: Let it be a regular polygon, and and be the bisectors of the angles, and (Fig. 150). Since, then, therefore, * 180°< 180°. Отсюда следует, что биссектрисы и углов и пересекаются в некоторой точке ABOUT. Let's prove that O = OA 2 = ABOUT =… = OA P . Triangle ABOUT isosceles, therefore ABOUT= ABOUT. According to the second criterion for the equality of triangles, therefore, ABOUT = ABOUT. Similarly, it is proved that ABOUT = ABOUT etc. So the point ABOUT is equidistant from all vertices of the polygon, so a circle with center ABOUT radius ABOUT is circumscribed about the polygon.

Let us now prove that there is only one circumscribed circle. Consider some three vertices of a polygon, for example, A 2 , . Since only one circle passes through these points, then around the polygon … You cannot describe more than one circle.

• 4 You can inscribe a circle into any regular polygon, and 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 circumscribed about a regular polygon coincides with the center of a circle inscribed in the same polygon.
• 7 Symmetry:

They say that a figure has symmetry (symmetrical) if there is such a movement (not identical) that translates this figure into itself.

• 7.1. A general triangle has no axes or centers of symmetry; it is asymmetrical. 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 them passing through its center. It also has rotational symmetry about the center with a rotation angle.

When 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 top and middle 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 does not have a center of symmetry.

8 Similarity:

With similarity and -gon goes into -gon, half-plane into half-plane, therefore convex n-the angle becomes convex n-gon.

Theorem: If the sides and angles of convex polygons 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 similarity coefficient.
• 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

Topic: polygons - 8th grade:

A line of adjacent segments that do not lie on the same straight line is called broken line.

The ends of the segments are peaks.

Each segment is link.

And all the sums of the lengths of the segments make up the total length broken line For example, AM + ME + EK + KO = length of the broken line

If the segments are closed, then this polygon(see above) .

The links in a polygon are called parties.

Sum of side lengths - perimeter polygon.

Vertices lying on one side are neighboring.

A segment connecting non-adjacent vertices is called diagonally.

Polygons called by number of sides: pentagon, hexagon, etc.

Everything inside the polygon is inner part of the plane, and everything that is outside - outer part of the plane.

Note! In the picture below- this is NOT a polygon, since there are additional common points on one line for non-adjacent segments.

Convex polygon lies on one side of each straight line. To determine it mentally (or with a drawing), we continue each side.

In a polygon as many angles as sides.

In a convex polygon sum of all interior angles equal to (n-2)*180°. n is the number of angles.

The polygon is called correct, if all its sides and angles are equal. So the calculation of its internal angles is carried out according to the formula (where n is the number of angles): 180° * (n-2) / n

Below are polygons, the sum of their angles and what one angle is equal to.

External angles of convex polygons are calculated as follows:

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Subject, student age: geometry, 9th grade

Purpose of the lesson: study types of polygons.

Educational task: to update, expand and generalize students’ knowledge about polygons; form an idea of ​​the “component parts” of a polygon; conduct a study of the number of constituent elements of regular polygons (from triangle to n-gon);

Developmental 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 assigned work, perseverance in achieving the goal.

During the classes: quote written on the board

“Nature speaks the language of mathematics, the letters of this language ... mathematical figures.” G.Galliley

At the beginning of the lesson, the class is divided into working groups (in our case, divided 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) awakening interest in the topic being studied, motivating each student for educational activities.

Technique: 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 in this family have “many angles”?

2. ... does a triangle belong to a large family of polygons, distinguished 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 limited by a closed broken line, which in turn can be simple, closed. Let's talk about the fact that polygons can be flat, regular, or convex. One of the flat polygons is a triangle, with which you have long been familiar (you can show students posters depicting polygons, a broken line, show their different types, you can also use TSO).

2. Conception stage

Goal: obtaining new information, understanding it, selecting it.

Technique: zigzag.

Forms of work: individual->pair->group.

Each member of the group is given a text on the topic of the lesson, and the text is compiled in such a way that it includes both information already known to the students and information that is completely new. Along with the text, students receive questions, the answers to which must be found in this text.

Polygons. Types of polygons.

Who hasn't heard about the mysterious Bermuda Triangle, in which 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, obtuse, rectangular), the triangle belongs to a large family of polygons, distinguished among many different geometric shapes on the plane.

The word “polygon” indicates that all figures in this family have “many angles.” 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 the segments connecting them A 1 A 2, A 2 A 3,.... 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 has no self-intersections (Fig. 2, 3).

A polyline 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 neighboring links do not lie on the same straight line (Fig. 5).

Substitute a specific number, for example 3, in the word “polygon” instead of the “many” part. You will get a triangle. Or 5. Then - a pentagon. Note that, as many angles as there are, there are as many sides, so these figures could well be called polylaterals.

The vertices of the broken line are called the vertices of the polygon, and the links of the broken line are called the sides of the polygon.

The polygon divides the plane into two areas: internal and external (Fig. 6).

A plane polygon or polygonal area is the finite part of a plane bounded by a polygon.

Two vertices of a polygon that are the ends of one side are called adjacent. Vertices that are not ends of one side are non-neighboring.

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, when connected to each other, can form other figures, which in turn are also polygons.

Segments connecting non-adjacent vertices of a polygon are called diagonals.

A polygon is called convex if it lies in the same half-plane relative 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 this vertex.

Let's prove the theorem (about the sum of the 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 valid. Let A 1 A 2 ...A 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 a polygon is the sum of the angles of all these triangles. The sum of the angles of each triangle is equal to 180 0, and the number of these triangles n is 2. Therefore, the sum of the angles of a convex n-gon A 1 A 2 ...A n is equal to 180 0 * (n - 2). The theorem is proven.

The exterior angle of a convex polygon at a given vertex is the angle adjacent to the interior angle of the polygon at this vertex.

A convex polygon is called regular if all its 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 craftsmen who decorated buildings. They made beautiful patterns, for example on parquet. But not all regular polygons could be used to make parquet. Parquet cannot be made from regular octagons. The fact is that each angle is equal to 135 0. And if some point is the vertex of two such octagons, then they will account for 270 0, and there is no place for the third octagon to fit there: 360 0 - 270 0 = 90 0. But for a square this is enough. Therefore, you can make parquet from regular octagons and squares.

The stars are also 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 called? What are the vertices of a polygon called? What are the sides of a polygon called?

2nd group

Which polygon is called flat? Give examples of polygons.

What is n – square?

Explain which vertices of a polygon are adjacent and which are not.

What is the diagonal of a polygon?

3 group

Which polygon is called convex?

Explain which angles of a polygon are external and which are internal?

Which polygon is called regular? 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 points, draw up a supporting summary, and present information in one of the graphic forms. Upon completion of work, students return to their work groups.

3. Reflection stage -

a) assessment of one’s knowledge, challenge to the next step of knowledge;

b) comprehension and appropriation of the information received.

Reception: research work.

Forms of work: individual->pair->group.

Working groups include specialists in answering each section of the proposed questions.

Returning to the working group, the expert introduces the answers to his questions to other group members. The group exchanges information between all members of the working group. Thus, in each working group, thanks to the work of experts, a general understanding of the topic being studied is formed.

Students' research work - filling out the table.

Solving interesting problems on the topic of the lesson.

• In a quadrilateral, draw a straight line so that it divides it into three triangles.
• How many sides does a regular polygon have, each of its interior angles measuring 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 equal to: 360 0, 380 0?

Summing up the lesson. Recording homework.

The part of the plane bounded by a closed broken line is called a polygon.

The segments of this broken line are called parties polygon. AB, BC, CD, DE, EA (Fig. 1) are the sides of the polygon ABCDE. The sum of all the sides of a polygon is called its perimeter.

The polygon is called convex, if it is located on one side of any of its sides, indefinitely extended beyond both vertices.

The MNPKO polygon (Fig. 1) will not be convex, since it is located on more than one side of the straight line KR.

We will only consider convex polygons.

The angles formed by two adjacent sides of a polygon are called its internal corners, and their tops are vertices of the polygon.

A straight line segment connecting two non-adjacent vertices of a polygon is called the diagonal of the polygon.

AC, AD - diagonals of the polygon (Fig. 2).

Angles adjacent to the interior angles of a polygon are called exterior angles of the polygon (Fig. 3).

Depending on the number of angles (sides), the polygon is called a triangle, quadrilateral, pentagon, etc.

Two polygons are said to be congruent if they can be brought together by overlapping.

Inscribed and circumscribed polygons

If all the vertices of a polygon lie on a circle, then the polygon is called inscribed into a circle, and the circle - described near the polygon (fig).

If all sides of a polygon are tangent to a circle, then the polygon is called described about a circle, and the circle is called inscribed into a polygon (Fig).

Similarity of polygons

Two polygons of the same name are called similar if the angles of one of them are respectively equal to the angles of the other, and the similar sides of the polygons are proportional.

Polygons with the same number of sides (angles) are called polygons of the same name.

The sides of similar polygons connecting the vertices of correspondingly equal angles are called similar (Fig).

So, for example, for the polygon ABCDE to be similar to the polygon A'B'C'D'E', it is necessary that: ∠A = ∠A' ∠B = ∠B' ∠C = ∠C' ∠D = ∠D' ∠ E = ∠E' and, in addition, AB / A'B' = BC / B'C' = CD / C'D' = DE / D'E' = EA / E'A' .

Ratio of perimeters of similar polygons

First, consider the property of a series of equal ratios. Let us, for example, have the following ratios: 2 / 1 = 4 / 2 = 6 / 3 = 8 / 4 =2.

Let's find the sum of the previous terms of these relations, then the sum of their subsequent terms and find the ratio of the resulting sums, we get:

$$ \frac(2 + 4 + 6 + 8)(1 + 2 + 3 + 4) = \frac(20)(10) = 2 $$

We get the same thing if we take a series of some other relations, for example: 2 / 3 = 4 / 6 = 6 / 9 = 8 / 12 = 10 / 15 = 2 / 3 Let’s find the sum of the previous terms of these relations and the sum of the subsequent ones, and then find the ratio of these sums, we get:

$$ \frac(2 + 4 + 5 + 8 + 10)(3 + 6 + 9 + 12 + 15) = \frac(30)(45) = \frac(2)(3) $$

In both cases, the sum of the previous members of a series of equal relations relates to the sum of subsequent members of the same series, just as the previous member of any of these relations relates to its subsequent one.

We derived this property by considering a number of numerical examples. It can be derived strictly and in a general form.

Now consider the ratio of the perimeters of similar polygons.

Let the polygon ABCDE be similar to the polygon A’B’C’D’E’ (Fig).

From the similarity of these polygons it follows that

AB / A’B’ = BC / B’C’ = CD / C’D’ = DE / D’E’ = EA / E’A’

Based on the property we derived for a series of equal ratios, we can write:

The sum of the previous terms of the relations we have taken represents the perimeter of the first polygon (P), and the sum of the subsequent terms of these relations represents the perimeter of the second polygon (P’), which means P / P’ = AB / A’B’.

Hence, The perimeters of similar polygons are related to their similar sides.

Ratio of areas of similar polygons

Let ABCDE and A’B’C’D’E’ be similar polygons (Fig).

It is known that ΔАВС ~ ΔA'В'С' ΔACD ~ ΔA'C'D' and ΔADE ~ ΔA'D'E'.

Besides,

;

Since the second ratios of these proportions are equal, which follows from the similarity of polygons, then

Using the property of a series of equal ratios we get:

Or

where S and S’ are the areas of these similar polygons.

Hence, The areas of similar polygons are related as the squares of similar sides.

The resulting formula can be converted to this form: S / S’ = (AB / A’B’) 2

Area of ​​an arbitrary polygon

Let it be necessary to calculate the area of ​​an arbitrary quadrilateral ABC (Fig.).

Let's draw a diagonal in it, for example AD. We get two triangles ABD and ACD, the areas of which we can calculate. Then we find the sum of the areas of these triangles. The resulting sum will express the area of ​​this quadrilateral.

If you need to calculate the area of ​​a pentagon, then we do the same thing: we draw diagonals from one of the vertices. We get three triangles, the areas of which we can calculate. This means we can find the area of ​​this pentagon. We do the same when calculating the area of ​​any polygon.

Projected area of ​​a polygon

Let us recall that the angle between a line and a plane is the angle between a given line and its projection onto the plane (Fig.).

Theorem. The area of ​​the orthogonal projection of a polygon onto a plane is equal to the area of ​​the projected polygon multiplied by the cosine of the angle formed by the plane of the polygon and the projection plane.

Each polygon can be divided into triangles whose sum of areas is equal to the area of ​​the polygon. Therefore, it is enough to prove the theorem for a triangle.

Let ΔАВС be projected onto the plane R. Let's consider two cases:

a) one of the sides ΔABC is parallel to the plane R;

b) none of the sides ΔABC are parallel R.

Let's consider first case: let [AB] || R.

Let us draw a plane through (AB) R 1 || R and project orthogonally ΔАВС on R 1 and on R(rice.); we get ΔАВС 1 and ΔА'В'С'.

By the property of projection we have ΔАВС 1 (cong) ΔА'В'С', and therefore

S Δ ABC1 = S Δ A’B’C’

Let's draw ⊥ and the segment D 1 C 1 . Then ⊥ , a \(\overbrace(CD_1C_1)\) = φ is the value of the angle between the plane ΔABC and the plane R 1 . That's why

S Δ ABC1 = 1 / 2 | AB | | C 1 D 1 | = 1 / 2 | AB | | CD 1 | cos φ = S Δ ABC cos φ

and therefore S Δ A’B’C’ = S Δ ABC cos φ.

Let's move on to consider second case. Let's draw a plane R 1 || R through that vertex ΔАВС, the distance from which to the plane R the smallest (let this be vertex A).

Let's project ΔАВС on the plane R 1 and R(rice.); let its projections be ΔАВ 1 С 1 and ΔА'В'С', respectively.

Let (BC) ∩ p 1 = D. Then

S Δ A’B’C’ = S ΔAB1 C1 = S ΔADC1 - S ΔADB1 = (S ΔADC - S ΔADB) cos φ = S Δ ABC cos φ