These geometric shapes are all around us.Convex polygons are natural, such as a honeycomb or artificial (man-made).These figures are used in the production of various types of coatings, painting, architecture, decoration, etc.Convex polygons have the property that all their points are on the same side of the line that passes through a pair of adjacent vertices of the geometric figure.There are other definitions.A convex polygon is called one, which is located in a single half-plane with respect to any line containing one of its sides.
convex polygons
The course of elementary geometry are always treated extremely simple polygons.To see all of the properties of geometric figures is necessary to understand their nature.To begin to understand that closed is any line whose ends are the same.And the figure formed by it, can have a variety of configurations.Polygon is called a simple closed polyline whose neighboring units are not located on the same line.Her links and nodes are respectively sides and vertices of the geometric figure.Simple polyline must not intersect itself.
neighboring vertices of the polygon are called, in the event that they are the ends of one of its sides.A geometric figure, which has a n-th number of vertices, and hence the n-th number of parties called the n-gon.Samu broken line called the border or contour of the geometric figure.Polygonal plane or flat polygon called the final part of any plane, they limited.Adjacent sides of the geometric figure called the broken line segments emanating from one vertex.They will not be neighbors if they are based on different vertices of the polygon.
Other definitions convex polygons
In elementary geometry, there are several equivalent in meaning definitions, indicating what is called a convex polygon.Moreover, all these statements are equally true.A convex polygon is the one that has:
• each segment that connects any two points within it, lies entirely in it;
• therein lie all its diagonals;
• any internal angle is less than 180 °.
Polygon always divides the plane into two parts.One of them - the limited (it can be enclosed in a circle), and the other - unlimited.The first is called the inner region, and the second - the outer region of the geometric figure.This is the intersection of the polygon (in other words - the common component) of several half-planes.In addition, each segment having ends at the points that belong to the polygon, is wholly owned by him.
Species convex polygons
definition of a convex polygon does not indicate that there are many kinds of them.And each of them has certain criteria.For convex polygons that have an internal angle of 180 °, called bulges slightly.Convex geometric figure that has three peaks, called a triangle, four - quadrangle, five - the pentagon, and so on. D. Each of the convex n-gon meets the following important requirements: n must be equal to or greater than 3. Each of the triangles is convex.The geometric figure of this type, in which all the vertices are on the same circle, called the inscribed circle.Described convex polygon is called if all its sides touch the circle around her.Two polygons called equal only in the case when using the overlay can be combined.Flat polygon is called a polygonal plane (of the plane), which is limited to this geometric figure.
regular convex polygons
regular polygons is called geometric shapes with equal angles and sides.Inside them there is a point 0, which is equidistant from each of its vertices.It is called the center of this geometric figure.Segment connecting the center with the vertices of the geometrical figure called apothem, and those that connect the point 0 with the parties - radii.
correct quadrangle - a square.The right triangle is called equilateral.For these figures there is the following rule: each corner of a convex polygon is 180 ° * (n-2) / n,
where n - the number of vertices of the convex geometry.
area of any regular polygon is determined by the formula:
S = p * h,
where p is equal to half the sum of all sides of the polygon, and h is the length of apothem.
Properties convex polygons
convex polygons have certain properties.Thus, a segment that connects any two points of a geometric figure, necessarily located therein.Proof:
assume that P - the convex polygon.Take two arbitrary points, such as A, B, which belong to P. By the current definition of a convex polygon, these points are located at one side of the straight line that contains any direction R. Consequently, AB also has this property and is contained in R. A convex polygon alwaysmay be divided into several triangles absolutely all diagonals who held one of its peaks.
convex angles of geometric shapes
angles of a convex polygon - the angles that are formed by the parties.The inner corners are in the inner area of the geometric figure.The angle that is formed by the parties, which meet at a vertex, called the angle of a convex polygon.The corners adjacent to the inner corners of the geometric figure, called external.Each corner of a convex polygon, located inside it is:
180 ° - x,
where x - the value of the outside corner.This simple formula is valid for any type of geometric shapes such.
In general, for the outer corners there is the following rule: each corner of a convex polygon is equal to the difference between 180 ° and the value of the inner corner.It can have values ranging from -180 ° to 180 °.Consequently, when the inner angle is 120 °, the appearance will have a value of 60 °.
sum of the angles of convex polygons
sum of the interior angles of a convex polygon is set by the formula:
180 ° * (n-2),
where n - the number of vertices of the n-gon.
sum of the angles of a convex polygon is calculated quite simply.Consider any such geometric shapes.To determine the sum of the angles in a convex polygon must be connected to one of its vertices to other vertices.As a result of this action turns (n-2) of the triangle.It is known that the sum of the angles of any triangle is always 180 °.Since the number in any polygon equals (n-2), the sum of the interior angles of the figure is equal to 180 ° x (n-2).
sum of the angles of a convex polygon, namely, any two inner and adjacent outer edges and at this convex geometric figure will always be equal to 180 °.On this basis, we can define the sum of all its angles:
180 x n.
sum of the interior angles of 180 ° * (n-2).Accordingly, the sum of all the outer corners of the figure is set by the formula:
180 ° * n-180 ° - (n-2) = 360 °.
sum of external angles of any convex polygon will always be equal to 360 ° (regardless of the number of its sides).Outside corner
convex polygon is generally represented by the difference between 180 ° and the value of internal angle.
Other properties of a convex polygon
addition to these basic properties of geometric figures, they also have others that arise when handling them.Thus, any of the polygons may be split into several convex n-gon.You must continue each of its sides and cut the geometric shape along these straight lines.Split any polygon into multiple convex portions and may be such that the tip of each of the pieces matched with all of its vertices.From a geometrical figure can be very simple to make triangles through all the diagonals from one vertex.Thus, any polygon, ultimately, can be divided into a certain number of triangles, which is very useful in solving various problems associated with these geometric shapes.
perimeter of a convex polygon
polyline segments, called sides of the polygon, often indicated by the following letters: ab, bc, cd, de, ea.This side of the geometric shapes with vertices a, b, c, d, e.The sum of the lengths of the sides of a convex polygon is called its perimeter.
circumference polygon
convex polygons can be inscribed and described.Circumference concerning all sides of the geometric figure called inscribed in it.This is called a polygon described.Center circle, which is inscribed in a polygon is the point of intersection of the bisectors of angles within a given geometric figure.The area of the polygon equals:
S = p * r,
where r - radius of the inscribed circle, and p - semiperimeter given polygon.
circle containing the vertices of the polygon described by him called.Furthermore, this convex geometric figure called the inscribed.Center circle described about this polygon is the point of intersection of the so-called midperpendiculars all sides.
diagonals of convex geometric shapes
diagonals of a convex polygon - a segment that connects neighboring vertices not.Each of them is inside the geometric shape.The number of diagonals of the n-gon is set according to the formula:
N = n (n - 3) / 2.
diagonal convex polygon number is important in elementary geometry.The number of triangles (R), which may break every convex polygon is calculated as follows:
K = n - 2.
number of diagonals of a convex polygon is always dependent on the number of vertices.
Splitting convex polygon
In some cases, to solve geometry tasks should be split into several convex polygon the triangles with disjoint diagonals.This problem can be solved by removing certain formula.
certain tasks: call the right kind of partition of a convex n-gon for several triangles diagonals intersect only at the vertices of a geometric figure.
Solution: Suppose that P1, P2, P3, ..., Pn - the top of this n-gon.Number Xn - the number of its partitions.Carefully look at the resulting diagonal geometric figure Pi Pn.In any of the correct partitions P1 Pn belongs to a particular triangle P1 Pi Pn, in which 1 & lt; i & lt; n.On this basis and assuming that i = 2,3,4 ..., n-1 is obtained (n-2) of these partitions, which include all possible special cases.
Let i = 2 is a group of regular partitions, always containing a diagonal P2 Pn.The number of partitions that are part of it, coincides with the number of partitions (n-1) -gon P2 P3 P4 ... Pn.In other words, it is equal to Xn-1.
If i = 3, then the other group partitions will always contain a diagonal P3 P1 and P3 Pn.The number of correct partitions that are contained in the group, will coincide with the number of partitions (n-2) -gon P3, P4 ... Pn.In other words, it will be Xn-2.
Let i = 4, then among triangles certainly correct partition will contain a triangle P4 P1 Pn, which will adjoin the quadrilateral P1 P2, P3, P4, (n-3) -gon P5 P4 ... Pn.The number of correct partitions such quadrilateral equals X4, and partition number (n-3) -gon equals Xn-3.Based on the foregoing, we can say that the total number of regular partitions that are contained in this group is equal to Xn-3 X4.Other groups that i = 4, 5, 6, 7 ... will contain Xn-4 X5, Xn-5 X6, Xn-6 X7 ... regular partitions.
Let i = n-2, the number of partitions in the right group is the same as the number of partitions in the group, in which i = 2 (in other words, equals Xn-1).
Since X1 = X2 = 0, X3 = 1, X4 = 2, ..., then the number of partitions of convex polygons equal:
Xn = Xn-1 + Xn-2 + Xn-3 + Xn-X4 X5 + 4 ...5 + X 4 + Xn-Xn-3 X4 + Xn-2 + Xn-1.
Example:
X5 = X4 + X3 + X4 = 5
X6 = X5 + X4 + X4 + X5 = 14
X7 = X6 + X5 + X4 * X4 + X5 + X6 = 42
X8 X7 =+ X6 + X5 * X4 + X4 * X5 + X6 + X7 = 132
correct number of partitions inside one diagonal cross
When testing special cases, it can be assumed that the number of diagonals of convex n-gon is equal to the product of all of the partitionsfigure to (n-3).
proof of this hypothesis: imagine that P1n = Xn * (n-3), then any n-gon may be divided into (n-2) a triangle.Moreover, from them can be stacked (n-3) -chetyrehugolnik.In addition, each quadrangle is diagonal.Since this convex geometric figure may be conducted two diagonals, which means that in all (n-3) may hold additional -chetyrehugolnikah diagonal (n-3).On this basis, we can conclude that in any right it is possible to carry out the partition (n-3) -diagonali that meet the conditions of this problem.
Area convex polygons
often in solving various problems of elementary geometry becomes necessary to determine the area of a convex polygon.Assume that (Xi. Yi), i = 1,2,3 ... n represents a sequence of coordinates of all the neighboring vertices of a polygon without self-intersections.In this case, its area is calculated by the following formula:
S = ½ (Σ (Xi + Xi + 1) (Yi + Yi + 1)),
where (X1, Y1) = (Xn +1, Yn + 1).
