A figure consisting of a polygon and n triangles. Geometric figure polygon. Collection and use of personal information

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) - 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, extended indefinitely beyond both vertices.

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

We will consider only convex polygons.

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

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

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

The corners adjacent to the internal corners of the polygon are called the external corners of the polygon (Fig. 3).

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

Two polygons are said to be equal if they can be superimposed.

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 around the 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 are called similar if they connect the vertices of correspondingly equal angles (Fig.).

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

Perimeter ratio of similar polygons

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

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

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

We will get the same if we take a number of some other relations, for example: 2 / 3 = 4 / 6 = 6 / 9 = 8 / 12 = 10 / 15 = 2 / 3 and then we 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 is related to the sum of subsequent members of the same series, as the previous member of any of these relations is related to its next one.

We deduced this property by considering a number of numerical examples. It can be deduced strictly and in 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.).

It follows from the similarity of these polygons that

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

Based on the property of a series of equal relations we have derived, we can write:

The sum of the previous terms of the relations we have taken is the perimeter of the first polygon (P), and the sum of the subsequent terms of these relations is the perimeter of the second polygon (P '), so P / P ' = AB / A'B '.

Consequently, the perimeters of similar polygons are related as their corresponding sides.

Ratio of areas of similar polygons

Let ABCDE and A'B'C'D'E' be similar polygons (fig.).

It is known that ΔABC ~ ΔA'B'C' 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.

Consequently, 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 required to calculate the area of ​​an arbitrary quadrilateral ABDC (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 ​​\u200b\u200bthe given quadrangle.

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

Polygon projection area

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 the polygon onto the 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, the sum of the areas of which is equal to the area of ​​the polygon. Therefore, it suffices to prove the theorem for a triangle.

Let ΔABC be projected onto the plane R. Consider two cases:

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

b) none of the sides ΔABC is parallel R.

Consider first case: let [AB] || R.

Draw through the (AB) plane R 1 || R and project orthogonally ΔABC onto R 1 and on R(rice.); we get ΔABC 1 and ΔA’B’C’.

By the projection property, we have ΔABC 1 (cong) ΔA’B’C’, and therefore

S ∆ ABC1 = S ∆ A'B'C'

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

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

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

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

Let's design ΔABC on the plane R 1 and R(rice.); let its projections be respectively ΔAB 1 C 1 and ΔA’B’C’.

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

§ 1 The concept of a triangle

In this lesson you will get acquainted with such shapes as a triangle and a polygon.

If three points that do not lie on the same straight line are connected by segments, then a triangle will be obtained. A triangle has three vertices and three sides.

Before you is a triangle ABC, it has three vertices (point A, point B and point C) and three sides (AB, AC and CB).

By the way, these same sides can be called differently:

AB=BA, AC=CA, CB=BC.

The sides of a triangle form three angles at the vertices of the triangle. In the picture you see angle A, angle B, angle C.

Thus, a triangle is a geometric figure formed by three segments that connect three points that do not lie on one straight line.

§ 2 The concept of a polygon and its types

In addition to triangles, there are quadrilaterals, pentagons, hexagons, and so on. In a word, they can be called polygons.

In the picture you see the DMKE quadrilateral.

Points D, M, K and E are the vertices of the quadrilateral.

The segments DM, MK, KE, ED are the sides of this quadrilateral. Just like in the case of a triangle, the sides of the quadrilateral form four corners at the vertices, you guessed it, hence the name - quadrilateral. For this quadrilateral, you see in the figure the D angle, the M angle, the K angle, and the E angle.

What quadrilaterals do you already know?

Square and Rectangle! Each of them has four corners and four sides.

Another type of polygon is a pentagon.

The points O, P, X, Y, T are the vertices of the pentagon, and the segments TO, OP, PX, XY, YT are the sides of this pentagon. A pentagon has five corners and five sides, respectively.

How many corners and how many sides do you think a hexagon has? That's right, six! Arguing in a similar way, we can say how many sides, vertices, or angles a particular polygon has. And we can conclude that a triangle is also a polygon, which has exactly three angles, three sides and three vertices.

Thus, in this lesson you got acquainted with such concepts as a triangle and a polygon. We learned that a triangle has 3 vertices, 3 sides and 3 angles, a quadrangle has 4 vertices, 4 sides and 4 angles, a pentagon has 5 sides, 5 vertices, 5 angles, respectively, and so on.

List of used literature:

1. Mathematics 5th grade. Vilenkin N.Ya., Zhokhov V.I. and others. 31st ed., ster. - M: 2013.
2. Didactic materials in mathematics Grade 5. Author - Popov M.A. - year 2013
3. We calculate without errors. Work with self-examination in mathematics grades 5-6. Author - Minaeva S.S. - year 2014
4. Didactic materials in mathematics Grade 5. Authors: Dorofeev G.V., Kuznetsova L.V. - 2010
5. Control and independent work in mathematics Grade 5. Authors - Popov M.A. - year 2012
6. Maths. Grade 5: textbook. for general education students. institutions / I. I. Zubareva, A. G. Mordkovich. - 9th ed., Sr. - M.: Mnemosyne, 2009

Knowledge of terminology, as well as knowledge of the properties of various geometric shapes, will help in solving many problems in geometry. Studying such a section as planimetry, the student often comes across the term “polygon”. What figure is characterized by this concept?

Polygon - definition of a geometric figure

A closed broken line, all sections of which lie in the same plane and do not have self-intersections, forms a geometric figure called a polygon. The number of links of the polyline must be at least 3. In other words, a polygon is defined as a part of a plane whose boundary is a closed broken line.

In the course of solving problems involving a polygon, concepts such as:

• The side of the polygon. This term characterizes a segment (link) of a broken chain of the desired figure.
• Polygon angle (internal) - the angle that is formed by 2 adjacent links of the polyline.
• The vertex of a polygon is defined as the vertex of a polyline.
• The diagonal of a polygon is a segment connecting any 2 vertices (except neighboring ones) of a polygonal figure.

In this case, the number of links and the number of vertices of the polyline within one polygon coincide. Depending on the number of corners (or segments of the broken line, respectively), the type of polygon is also determined:

• 3 corners - triangle.
• 4 corners - quadrilateral.
• 5 corners - pentagon, etc.

If a polygonal figure has equal angles and, accordingly, sides, then they say that this polygon is regular.

Types of polygons

All polygonal geometric shapes are divided into 2 types - convex and concave.

• If any of the sides of the polygon, after continuing to a straight line, does not form intersection points with the actual figure, then you have a convex polygonal figure.
• If, after the extension of the (any) side, the resulting line intersects the polygon, we are talking about a concave polygon.

Polygon Properties

Regardless of whether the studied polygonal figure is regular or not, it has the following properties. So:

• Its internal angles form (p – 2)*π in total, where

π is the radian measure of a straightened angle, corresponds to 180°,

p is the number of corners (vertices) of a polygonal figure (p-gon).

• The number of diagonals of any polygonal figure is determined from the ratio p * (p - 3) / 2, where

p is the number of sides of the p-gon.

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 (symmetric) 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

Types of polygons:

Quadrangles

Quadrangles, respectively, consist of 4 sides and corners.

Sides and angles that are opposite each other are called opposite.

Diagonals divide convex quadrilaterals into triangles (see figure).

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 (numbered 1 in the figure).

Opposite sides and angles in a parallelogram are always equal.

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

Trapeze

Trapeze is also a quadrilateral, and trapeze only two sides are parallel, which are called grounds. The other sides are sides.

The trapezoid in the figure is numbered 2 and 7.

As in the 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 a trapezoid is half the sum of the bases and parallel to them.

Rhombus

Rhombus is a parallelogram with all sides 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 figure, the rhombus is numbered 5.

Rectangles

Rectangle- this is a parallelogram, in which each corner is a right one (see in the figure at 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 (#4).

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