Polygons. Visual Guide (2019)

The vertices of the polygon, and the segments are the sides of the polygon. Vertices of a polygon - page No. 1/1

Geometry 8th grade K.K.Kurginyan Part-1* (with an asterisk).
Polygon.

Definition: A polygon is a geometric figure that consists of a flat, closed broken line without self-intersections. The vertices of the broken line are called peaks polygon, and the segments are parties polygon.

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

External corner 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 external angle is the difference between 180° and an internal angle; it can take values from -180° to 180°. The sum of the external angles of a polygon is 360°.

Convex polygon.
Polygonis called convex if:
DefinitionI - for any two points inside it, the segment connecting them lies entirely in it.

DefinitionII - each internal angle is less than 180°.

DefinitionIII - all its diagonals lie completely inside it.

DefinitionIV – it lies on one side of each straight line passing through its two neighboring vertices.
Sum of angles n -gon.
The sum of the angles of a convex n-gon is (n-2)∙180°.
The sum of the angles of a non-convex n-gon is also equal to (n-2)∙180°. (The proof is similar, but uses in addition the lemma that any polygon can be cut diagonally into triangles).
Number of diagonals n -gon.*

Theorem: The number of diagonals of any n-gon is n(n-3)2.

Proof: Let n be the number of vertices of the polygon, let us calculate p the number of possible different diagonals. Each vertex is connected by diagonals to all other vertices, except the two neighboring ones and, naturally, itself. Thus, n-3 diagonals can be drawn from one vertex; Let's multiply this by the number of vertices (n-3)∙n, however, we counted each diagonal twice (once for each end, therefore, we must divide by 2) - hence, p= n(n-3)2.

Problem*: Which convex polygon has 25 more diagonals than sides?

25+n = nn-32

50 + 2n = n 2 - 3n

n 2 - 5n - 50 = 0

Let's factorize

n 2 -25-5n -25 = 0

n=-5 does not satisfy,

because it doesn't exist

such a polygon

n = 10 satisfies

Answer: Decagon.

Shapes with equal diagonals.*

On surface there are two regular polygons with all diagonals are equal among themselves - this square And regular pentagon (pentagon). A square has two identical diagonals that intersect at a right angle in the center. A regular pentagon has five identical diagonals, which together form the pattern of a five-pointed star (pentagram).

In space there is only one correct one polyhedron (not a polygon), which one all diagonals are equal among themselves - this regular octahedron (octahedron). At the octahedron three diagonals that intersect in pairs perpendicularly in the center. All diagonals of the octahedron are spatial (the octahedron has no diagonals of faces, since it has triangular faces).

In addition to the octahedron, there is another regular polyhedron, which all spatial diagonals are equal among themselves - this cube (hexahedron), in addition to spatial ones, the cube has diagonals of faces. The cube has four identical spatial diagonals that intersect in the center. The angle between the diagonals of a cube is either arccos (1/3) ≈ 70.5° (for a pair of diagonals drawn to adjacent vertices) or arccos (–1/3) ≈ 109.5° (for a pair of diagonals drawn to non-adjacent vertices ).

Quadrilaterals.
Each quadrilateral has four vertices, four sides and two diagonals.

Two non-adjacent sides are called opposite sides.

Two non-adjacent vertices are called opposite.
1.Parallelogram is a quadrilateral whose opposite sides are parallel in pairs.
Properties of a parallelogram:
1) Opposite sides of a parallelogram are equal. AB=DC, AD=BC.

2) Opposite angles of a parallelogram are equal. A=C, B=D.

3) The diagonals of a parallelogram intersect and are divided in half by the intersection point. AO=OC, BO=OD.

4) The sum of angles adjacent to one side is 180°. A+D=180, A+B=180, B+C=180, D+C=180.

5) The sum of all angles is 360°. A+B+C+D=360°.

6)* The sum of the squares of the diagonals of a parallelogram is equal to twice the sum of the squares of its two adjacent sides: AC 2 +BD 2 =2∙(AB 2 +AD 2).

Problem 1*: Find the diagonal of a parallelogram if it is known that the length of one diagonal is AC = 9 cm, and the sides AD = 7 cm and AB = 4 cm.

Solution: Substituting the values into the formula we get:

81+BD 2 =2∙(49+16),

BD 2 =49, therefore the second diagonal is BD = 7 cm. Answer: 7 cm.
Problem 2*: Find the diagonal of a parallelogram if it is known that the length of one diagonal is BD=10 cm, and the sides AD=8 cm and AB=2 cm.

Solution: The conditions of the problem are not true, since the sum of two sides of a triangle is always greater than the third side. Answer: the problem has no solutions (meaning).

Problem 3*: a) Find the side of the parallelogram if it is known that the length of the diagonals is BD = 6 cm, AC = 8, and one side AB = 5 cm. b) What is the name of this parallelogram.
Problem 4**: The sum of the lengths of the diagonals of a parallelogram is 12 cm, and the product of 32, find the value of the sum of the squares of all its sides.
Problem 5**: Find the largest perimeter of a parallelogram whose diagonals are 6 cm and 8 cm.

Solution: Let's prove that among all parallelograms with given diagonal lengths, the rhombus has the largest perimeter .

Indeed, let a And b are the lengths of adjacent sides of the parallelogram, and and are the lengths of its diagonals (see Fig. 2). Then the perimeter of the parallelogram is: P = 2(a + b).

From the equality expressing the theorem on the sum of the squares of the diagonals of a parallelogram, it follows that for all parallelograms with given diagonals, the sum of the squares of the sides is a constant value.

According to the inequality between the arithmetic mean and the mean square:  , and equality is achieved t. and t. t., when a = b. This means that the parallelogram with the largest perimeter is a rhombus. Find the side of this rhombus: =5(cm). Answer: 20 cm.

2.Rectangle is a parallelogram in which all angles are right.
Definition 2: This is a quadrilateral with all right angles.

Definition 3: It is a parallelogram with one right angle.

Definition 4: It is a parallelogram whose angles are equal.
Rectangle properties: +
1) The diagonals of the rectangle are equal.

2)* The square of the diagonal is equal to the sum of the squares of the sides. AC 2 =AB 2 +DC 2

Task 1: The shortest side of the rectangle is 5cm, the diagonals intersect at an angle of 60°. Find the diagonals of the rectangle.
Task 2: The shortest side of a rectangle is 24, the diagonals intersect at an angle of 120°. Find the diagonals and the longest side of the rectangle.
Problem 3*: The side of the rectangle is 3 cm, the diagonal is 5 cm. Find the other side of the rectangle.
Problem 4*: The side of the rectangle is 6 cm, the diagonal is 10 cm. Find the area of the rectangle.

3.Rhombus is a parallelogram in which all sides are equal.
Definition 2: It is a quadrilateral with all sides equal.
Properties of a rhombus: same properties as a parallelogram +
1) The diagonals of a rhombus are mutually perpendicular (AC ⊥ BD).

2) The diagonals of a rhombus divide its angles in half (that is, the diagonals of a rhombus are the bisectors of its angles - ∠DCA = ∠BCA, ∠ABD = ∠CBD, ∠BAC = ∠DAC, ∠ADB = ∠CDB).

3)*The sum of the squares of the diagonals is equal to the square of the side multiplied by 4 (a consequence of the parallelogram identity). AC 2 +BD 2 =4 AB 2
Task 1: The diagonals of the rhombus are 6 and 8 cm. Find the side of the rhombus.
Task 2: The side of the rhombus is 10 cm, one of the angles is 60. Find the small diagonal of the rhombus.
4.Square is a parallelogram in which all angles are equal to 90 and all sides are equal.
Definition 2: This is a parallelogram in which all angles and sides are equal to each other.

Definition 3: This is a quadrilateral in which all angles and sides are equal to each other.

Definition 4: This is a rhombus with one right angle.

Definition 5: This is a rhombus whose angles are equal.

Definition 6: It is a rectangle with all sides equal.
Properties of a square: same properties as a parallelogram +
1) The diagonals of a square are equal.

2) The diagonals of the square are mutually perpendicular (AC ⊥ BD).

3) The diagonals of a square divide its angles in half (that is, the diagonals of a square are the bisectors of its angles - ∠DCA = ∠BCA= ∠ABD = ∠CBD= ∠BAC = ∠DAC= ∠ADB = ∠CDB=45).

4)* The square of the diagonal is equal to twice the square of the side. AC 2 =2 AB 2

5.Trapezoid is a quadrilateral in which two sides are parallel and the other two are not parallel.
The parallel sides are called bases, and the other two are called lateral sides.

A trapezoid is called isosceles if its sides are equal.

A trapezoid is called rectangular if one of its angles is right.
Task: Prove that a trapezoid cannot be both rectangular and isosceles.

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. For > 3, each vertex of the -gon has 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 straight 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 using 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:

There are different points of view on what is considered a polygon. In a school geometry course, one of the following definitions is used.

Definition 1

Polygon

is a figure made up of segments

so that adjacent segments(that is, adjacent segments with a common vertex, for example, A1A2 and A2A3) do not lie on the same line, and non-adjacent segments do not have common points.

Definition 2

A simple closed polygon is called a polygon.

Points

are called vertices of the polygon, segments

— sides of the polygon.

The sum of the lengths of all sides is called perimeter of the polygon.

A polygon that has n vertices (and therefore n sides) is called n - square.

A polygon that lies in the same plane is called flat. When people talk about a polygon, unless otherwise stated, they mean a flat polygon.

Two vertices belonging to the same side of a polygon are called neighboring. For example, A1 and A2, A5 and A6 are neighboring vertices.

A segment that connects two non-adjacent vertices is called polygon diagonal.

Let's find out how many diagonals a polygon has.

From each of the n vertices of the polygon there are n-3 diagonals

(there are n vertices in total. We do not count the vertex itself and two neighboring vertices that do not form a diagonal with this vertex. For vertex A1, for example, we do not take into account A1 itself and the neighboring vertices A2 and A3).

Thus, each of the n vertices corresponds to n-3 diagonals. Since one diagonal refers to two vertices at once, in order to find the number of diagonals of a polygon, the product n(n-3) must be divided in half.

Therefore, n - a triangle has

diagonals.

Any polygon divides the plane into two parts - the internal and external regions of the polygon. A figure consisting of a polygon and its interior region is also called a polygon.

Definition

Top corner

The vertex of the angle is the point where the two rays originate.

The vertex of the angle is the point where the two rays originate; where two segments meet; where two lines intersect; where is any combination of rays, segments and lines that form two (straight-line) “sides” that converge at one point.

Polygon polygon vertex

In a polygon, a vertex is called "convex" if the interior angle of the polygon is less than π radians (180° is two right angles). Otherwise the vertex is called "concave".

More generally, a vertex of a polyhedron is convex if the intersection of the polyhedron with a sufficiently small sphere having the vertex as its center is a convex figure; otherwise the apex is concave.

The vertices of a polyhedron are related to the vertices of a graph, since a polyhedron is a graph whose vertices correspond to the vertices of a polyhedron, and therefore the graph of a polyhedron can be considered as a one-dimensional simplicial complex whose vertices are the vertices of the graph. However, in graph theory, vertices can have fewer than two incident edges, which is usually not allowed for geometric vertices. There is also a connection between geometric vertices and the vertices of the curve, the points of extrema of its curvature - the vertices of the polygon are in a sense points of infinite curvature, and if the polygon is approximated by a smooth curve, the points of extreme curvature will lie near the vertices of the polygon. However, approximating the polygon using a smooth curve gives additional vertices at the points of minimum curvature.

Vertices of flat mosaics

"Ears"

"Mouths"

Main peak x i (\displaystyle x_(i)) simple polygon P (\displaystyle P) called "mouth" if the diagonal [ x i − 1 , x i + 1 ] (\displaystyle ) lies outside P (\displaystyle P).

Number of vertices of a polyhedron

Any surface of a three-dimensional convex polyhedron has the Euler characteristic:

V − E + F = 2 , (\displaystyle V-E+F=2,)

Where V (\displaystyle V)- number of vertices, E (\displaystyle E)- number of edges, and F (\displaystyle F)- number of faces. This equality is known as Euler's equation. For example, a cube has 12 edges and 6 faces, and therefore 8 vertices: 8 − 12 + 6 = 2 (\displaystyle 8-12+6=2) .

Peaks in computer graphics

In computer graphics, objects are often represented as triangulated polyhedra, in which the vertices of an object are associated not only with three spatial coordinates, but also with other graphic information necessary for the correct construction of an image of an object, such as color, reflectivity, texture, and vertex normals. These properties are used when constructing an image using