It lies at the base of the parallelepiped. Types of parallelepiped

In this lesson we will define a parallelepiped, discuss its structure and its elements (diagonals of a parallelepiped, sides of a parallelepiped and their properties). We will also consider the properties of the faces and diagonals of a parallelogram. Next, we will solve a typical problem of constructing a section in a parallelepiped.

Topic: Parallelism of lines and planes

Lesson: Parallelepiped. Properties of faces and diagonals of a parallelepiped

In this lesson we will define a parallelepiped, discuss its structure, properties and its elements (sides, diagonals).

The parallelepiped is formed using two equal parallelograms ABCD and A 1 B 1 C 1 D 1, which are in parallel planes. Designation: ABCDA 1 B 1 C 1 D 1 or AD 1 (Fig. 1.).

2. Festival of pedagogical ideas "Open Lesson" ()

1. Geometry. Grades 10-11: textbook for students of general education institutions (basic and specialized levels) / I. M. Smirnova, V. A. Smirnov. - 5th edition, corrected and expanded - M.: Mnemosyne, 2008. - 288 pp.: ill.

Tasks 10, 11, 12 p. 50

2. Construct a section of a rectangular parallelepiped ABCDA1B1C1D1 plane passing through the points:

a) A, C, B1

b) B1, D1 and the middle of the rib AA1.

3. The edge of the cube is equal to a. Construct a section of the cube with a plane passing through the midpoints of three edges emerging from one vertex, and calculate its perimeter and area.

4. What shapes can be obtained as a result of the intersection of a parallelepiped plane?

In this lesson, everyone will be able to study the topic “Rectangular parallelepiped”. At the beginning of the lesson, we will repeat what arbitrary and straight parallelepipeds are, remember the properties of their opposite faces and diagonals of the parallelepiped. Then we'll look at what a cuboid is and discuss its basic properties.

Topic: Perpendicularity of lines and planes

Lesson: Cuboid

A surface composed of two equal parallelograms ABCD and A 1 B 1 C 1 D 1 and four parallelograms ABV 1 A 1, BCC 1 B 1, CDD 1 C 1, DAA 1 D 1 is called parallelepiped(Fig. 1).

Rice. 1 Parallelepiped

That is: we have two equal parallelograms ABCD and A 1 B 1 C 1 D 1 (bases), they lie in parallel planes so that the side edges AA 1, BB 1, DD 1, CC 1 are parallel. Thus, a surface composed of parallelograms is called parallelepiped.

Thus, the surface of a parallelepiped is the sum of all the parallelograms that make up the parallelepiped.

1. The opposite faces of a parallelepiped are parallel and equal.

(the shapes are equal, that is, they can be combined by overlapping)

For example:

ABCD = A 1 B 1 C 1 D 1 (equal parallelograms by definition),

AA 1 B 1 B = DD 1 C 1 C (since AA 1 B 1 B and DD 1 C 1 C are opposite faces of the parallelepiped),

AA 1 D 1 D = BB 1 C 1 C (since AA 1 D 1 D and BB 1 C 1 C are opposite faces of the parallelepiped).

2. The diagonals of a parallelepiped intersect at one point and are bisected by this point.

The diagonals of the parallelepiped AC 1, B 1 D, A 1 C, D 1 B intersect at one point O, and each diagonal is divided in half by this point (Fig. 2).

Rice. 2 The diagonals of a parallelepiped intersect and are divided in half by the intersection point.

3. There are three quadruples of equal and parallel edges of a parallelepiped: 1 - AB, A 1 B 1, D 1 C 1, DC, 2 - AD, A 1 D 1, B 1 C 1, BC, 3 - AA 1, BB 1, CC 1, DD 1.

Definition. A parallelepiped is called straight if its lateral edges are perpendicular to the bases.

Let the side edge AA 1 be perpendicular to the base (Fig. 3). This means that straight line AA 1 is perpendicular to straight lines AD and AB, which lie in the plane of the base. This means that the side faces contain rectangles. And the bases contain arbitrary parallelograms. Let us denote ∠BAD = φ, the angle φ can be any.

Rice. 3 Right parallelepiped

So, a right parallelepiped is a parallelepiped in which the side edges are perpendicular to the bases of the parallelepiped.

Definition. The parallelepiped is called rectangular, if its lateral edges are perpendicular to the base. The bases are rectangles.

The parallelepiped ABCDA 1 B 1 C 1 D 1 is rectangular (Fig. 4), if:

1. AA 1 ⊥ ABCD (lateral edge perpendicular to the plane of the base, that is, a straight parallelepiped).

2. ∠BAD = 90°, i.e. the base is a rectangle.

Rice. 4 Rectangular parallelepiped

A rectangular parallelepiped has all the properties of an arbitrary parallelepiped. But there are additional properties that are derived from the definition of a cuboid.

So, cuboid is a parallelepiped whose side edges are perpendicular to the base. The base of a cuboid is a rectangle.

1. In a rectangular parallelepiped, all six faces are rectangles.

ABCD and A 1 B 1 C 1 D 1 are rectangles by definition.

2. Lateral ribs are perpendicular to the base. This means that all the lateral faces of a rectangular parallelepiped are rectangles.

3. All dihedral angles of a rectangular parallelepiped are right.

Let us consider, for example, the dihedral angle of a rectangular parallelepiped with edge AB, i.e., the dihedral angle between planes ABC 1 and ABC.

AB is an edge, point A 1 lies in one plane - in the plane ABB 1, and point D in the other - in the plane A 1 B 1 C 1 D 1. Then the dihedral angle under consideration can also be denoted as follows: ∠A 1 ABD.

Let's take point A on edge AB. AA 1 is perpendicular to edge AB in the plane АВВ-1, AD is perpendicular to edge AB in the plane ABC. This means that ∠A 1 AD is the linear angle of a given dihedral angle. ∠A 1 AD = 90°, which means that the dihedral angle at edge AB is 90°.

∠(ABB 1, ABC) = ∠(AB) = ∠A 1 ABD= ∠A 1 AD = 90°.

Similarly, it is proved that any dihedral angles of a rectangular parallelepiped are right.

The square of the diagonal of a rectangular parallelepiped is equal to the sum of the squares of its three dimensions.

Note. The lengths of the three edges emanating from one vertex of a cuboid are the measurements of the cuboid. They are sometimes called length, width, height.

Given: ABCDA 1 B 1 C 1 D 1 - rectangular parallelepiped (Fig. 5).

Prove: .

Rice. 5 Rectangular parallelepiped

Proof:

Straight line CC 1 is perpendicular to plane ABC, and therefore to straight line AC. This means that the triangle CC 1 A is right-angled. According to the Pythagorean theorem:

Consider the right triangle ABC. According to the Pythagorean theorem:

But BC and AD are opposite sides of the rectangle. So BC = AD. Then:

Because , A , That. Since CC 1 = AA 1, this is what needed to be proven.

The diagonals of a rectangular parallelepiped are equal.

Let us denote the dimensions of the parallelepiped ABC as a, b, c (see Fig. 6), then AC 1 = CA 1 = B 1 D = DB 1 =

Translated from Greek, parallelogram means plane. A parallelepiped is a prism with a parallelogram at its base. There are five types of parallelogram: oblique, straight and cuboid. The cube and rhombohedron also belong to the parallelepiped and are its variety.

Before moving on to the basic concepts, let's give some definitions:

The diagonal of a parallelepiped is a segment that unites the vertices of the parallelepiped that are opposite each other.
If two faces have a common edge, then we can call them adjacent edges. If there is no common edge, then the faces are called opposite.
Two vertices that do not lie on the same face are called opposite.

What properties does a parallelepiped have?

The faces of a parallelepiped lying on opposite sides are parallel to each other and equal to each other.
If you draw diagonals from one vertex to another, then the point of intersection of these diagonals will divide them in half.
The sides of the parallelepiped lying at the same angle to the base will be equal. In other words, the angles of the co-directed sides will be equal to each other.

What types of parallelepiped are there?

Now let's figure out what kind of parallelepipeds there are. As mentioned above, there are several types of this figure: straight, rectangular, inclined parallelepiped, as well as cube and rhombohedron. How do they differ from each other? It's all about the planes that form them and the angles they form.

Let's look in more detail at each of the listed types of parallelepiped.

As is already clear from the name, an inclined parallelepiped has inclined faces, namely those faces that are not at an angle of 90 degrees in relation to the base.
But for a right parallelepiped, the angle between the base and the edge is exactly ninety degrees. It is for this reason that this type of parallelepiped has such a name.
If all the faces of the parallelepiped are identical squares, then this figure can be considered a cube.
A rectangular parallelepiped received this name because of the planes that form it. If they are all rectangles (including the base), then this is a cuboid. This type of parallelepiped is not found very often. Translated from Greek, rhombohedron means face or base. This is the name given to a three-dimensional figure whose faces are rhombuses.

Basic formulas for a parallelepiped

The volume of a parallelepiped is equal to the product of the area of the base and its height perpendicular to the base.

The area of the lateral surface will be equal to the product of the perimeter of the base and the height.
Knowing the basic definitions and formulas, you can calculate the base area and volume. The base can be chosen at your discretion. However, as a rule, a rectangle is used as the base.

There are several types of parallelepipeds:

· Rectangular parallelepiped- is a parallelepiped, all of whose faces are - rectangles;

· A right parallelepiped is a parallelepiped that has 4 side faces - parallelograms;

· An inclined parallelepiped is a parallelepiped whose side faces are not perpendicular to the bases.

Essential elements

Two faces of a parallelepiped that do not have a common edge are called opposite, and those that have a common edge are called adjacent. Two vertices of a parallelepiped that do not belong to the same face are called opposite. Line segment, connecting opposite vertices is called diagonally parallelepiped. The lengths of three edges of a rectangular parallelepiped having a common vertex are called measurements.

Properties

· The parallelepiped is symmetrical about the middle of its diagonal.

· Any segment with ends belonging to the surface of the parallelepiped and passing through the middle of its diagonal is divided in half by it; in particular, all diagonals of a parallelepiped intersect at one point and are bisected by it.

· Opposite faces of a parallelepiped are parallel and equal.

· The square of the diagonal length of a rectangular parallelepiped is equal to the sum of the squares of its three dimensions

Basic formulas

Right parallelepiped

· Lateral surface area S b =P o *h, where P o is the perimeter of the base, h is the height

· Total surface area S p =S b +2S o, where S o is the base area

· Volume V=S o *h

Rectangular parallelepiped

· Lateral surface area S b =2c(a+b), where a, b are the sides of the base, c is the side edge of the rectangular parallelepiped

· Total surface area S p =2(ab+bc+ac)

· Volume V=abc, where a, b, c are the dimensions of a rectangular parallelepiped.

· Lateral surface area S=6*h 2, where h is the height of the cube edge

34. Tetrahedron- regular polyhedron, has 4 faces that are regular triangles. Vertices of a tetrahedron 4 , converges to each vertex 3 ribs, and total ribs 6 . Also, a tetrahedron is a pyramid.

The triangles that make up a tetrahedron are called faces (AOS, OSV, ACB, AOB), their sides --- ribs (AO, OC, OB), and the vertices --- vertices (A, B, C, O) tetrahedron. Two edges of a tetrahedron that do not have common vertices are called opposite... Sometimes one of the faces of the tetrahedron is isolated and called basis, and the other three --- side faces.

The tetrahedron is called correct, if all its faces are equilateral triangles. Moreover, a regular tetrahedron and a regular triangular pyramid are not the same thing.

U regular tetrahedron all dihedral angles at the edges and all trihedral angles at the vertices are equal.

35. Correct prism

A prism is a polyhedron whose two faces (bases) lie in parallel planes, and all the edges outside these faces are parallel to each other. The faces other than the bases are called side faces, and their edges are called side edges. All side edges are equal to each other as parallel segments bounded by two parallel planes. All lateral faces of the prism are parallelograms. The corresponding sides of the bases of the prism are equal and parallel. A prism whose side edge is perpendicular to the plane of the base is called a straight prism; other prisms are called inclined. At the base of a regular prism is a regular polygon. All faces of such a prism are equal rectangles.

The surface of the prism consists of two bases and a side surface. The height of a prism is a segment that is a common perpendicular to the planes in which the bases of the prism lie. The height of the prism is the distance H between the planes of the bases.

Lateral surface area S b of a prism is the sum of the areas of its lateral faces. Total surface area S n of a prism is the sum of the areas of all its faces. S n = S b + 2 S,Where S– area of the base of the prism, S b – lateral surface area.

36. A polyhedron that has one face, called basis, – polygon,
and the other faces are triangles with a common vertex, called pyramid .

Faces other than the base are called lateral.
The common vertex of the lateral faces is called the top of the pyramid.
The edges connecting the top of the pyramid with the vertices of the base are called lateral.
Pyramid height is called a perpendicular drawn from the top of the pyramid to its base.

The pyramid is called correct, if its base is a regular polygon and its height passes through the center of the base.

Apotheme the lateral face of a regular pyramid is the height of this face drawn from the vertex of the pyramid.

A plane parallel to the base of the pyramid cuts it off into a similar pyramid and truncated pyramid.

Properties of regular pyramids

The lateral edges of a regular pyramid are equal.
The lateral faces of a regular pyramid are isosceles triangles equal to each other.

If all side edges are equal, then

·height is projected to the center of the circumscribed circle;

The side ribs form equal angles with the plane of the base.

If the side faces are inclined to the plane of the base at the same angle, then

·height is projected to the center of the inscribed circle;

· the heights of the side faces are equal;

·the area of the side surface is equal to half the product of the perimeter of the base and the height of the side face

37. The function y=f(x), where x belongs to the set of natural numbers, is called a function of a natural argument or a number sequence. It is denoted by y=f(n), or (y n)

Sequences can be specified in various ways, verbally, this is how a sequence of prime numbers is specified:

2, 3, 5, 7, 11, etc.

A sequence is considered to be given analytically if the formula for its nth term is given:

1, 4, 9, 16, …, n 2, …

2) y n = C. Such a sequence is called constant or stationary. For example:

2, 2, 2, 2, …, 2, …

3) y n =2 n . For example,

2, 2 2, 2 3, 2 4, …, 2 n, …

A sequence is said to be bounded above if all its terms are not greater than a certain number. In other words, a sequence can be called bounded if there is a number M such that the inequality y n is less than or equal to M. The number M is called the upper bound of the sequence. For example, the sequence: -1, -4, -9, -16, ..., - n 2 ; limited from above.

Similarly, a sequence can be called bounded below if all its terms are greater than a certain number. If a sequence is bounded both above and below it is called bounded.

A sequence is called increasing if each subsequent term is greater than the previous one.

A sequence is called decreasing if each subsequent member is less than the previous one. Increasing and decreasing sequences are defined by one term - monotonic sequences.

Consider two sequences:

1) y n: 1, 3, 5, 7, 9, …, 2n-1, …

2) x n: 1, ½, 1/3, 1/ 4, …, 1/n, …

If we depict the terms of this sequence on the number line, we will notice that, in the second case, the terms of the sequence are condensed around one point, but in the first case this is not the case. In such cases, the sequence y n is said to diverge and the sequence x n to converge.

The number b is called the limit of the sequence y n if any pre-selected neighborhood of the point b contains all members of the sequence, starting from a certain number.

In this case we can write:

If the quotient of a progression is less than one in modulus, then the limit of this sequence, as x tends to infinity, is equal to zero.

If the sequence converges, then only to one limit

If the sequence converges, then it is bounded.

Weierstrass's theorem: If a sequence converges monotonically, then it is bounded.

The limit of a stationary sequence is equal to any term of the sequence.

Properties:

1) The amount limit is equal to the sum of the limits

2) The limit of a product is equal to the product of the limits

3) The limit of the quotient is equal to the quotient of the limits

4) The constant factor can be taken beyond the limit sign

Question 38
sum of infinite geometric progression

Geometric progression- a sequence of numbers b 1, b 2, b 3,.. (members of the progression), in which each subsequent number, starting from the second, is obtained from the previous one by multiplying it by a certain number q (denominator of the progression), where b 1 ≠0, q ≠0.

Sum of an infinite geometric progression is the limiting number to which the sequence of progression converges.

In other words, no matter how long a geometric progression is, the sum of its terms is not more than a certain number and is practically equal to this number. This is called the sum of a geometric progression.

Not every geometric progression has such a limiting sum. It can only be for a progression whose denominator is a fractional number less than 1.

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