Projections of the diagonal vector of a parallelepiped onto its edges. Parallelepiped and cube. Visual Guide (2019). Protection of personal information

CHAPTER THREE

Polytopes

1. PARALLELEPIPED AND PYRAMID

Box face and diagonal properties

72. Theorem. In a parallelepiped:

1)opposite faces are equal and parallel;

2) all four diagonals intersect at one point and are divided in half at it.

1) The faces (Fig. 80) BB 1 C 1 C and AA 1 D 1 D are parallel, because two intersecting lines BB 1 and B 1 C 1 of one face are parallel to two intersecting lines AA 1 and A 1 D 1 of the other (§ 15 ); these faces and are equal, since B 1 C 1 = A 1 D 1, B 1 B = A 1 A (as opposite sides of parallelograms) and / BB 1 C 1 = / AA 1 D 1.

2) Take (Fig. 81) any two diagonals, for example AC 1 and BD 1, and draw auxiliary lines AD 1 and BC 1.

Since the edges AB and D 1 C 1 are respectively equal and parallel to the edge DC, they are equal and parallel to each other; as a result, the figure AD 1 C 1 B is a parallelogram, in which the straight lines C 1 A and BD 1 are diagonals, and in the parallelogram the diagonals are divided in half at the point of intersection.

Let us now take one of these diagonals, for example AC 1, with the third diagonal, let us assume with B 1 D. In the same way, we can prove that they are divided in half at the point of intersection. Consequently, the diagonals B 1 D and AC 1 and the diagonals AC 1 and BD 1 (which we took earlier) intersect at the same point, precisely in the middle of the diagonal
AC 1. Finally, taking the same diagonal AC 1 with the fourth diagonal A 1 C, we also prove that they are halved. Hence, the point of intersection of this pair of diagonals lies in the middle of the diagonal AC 1. Thus, all four diagonals of the parallelepiped intersect at the same point and are halved by this point.

73. Theorem. In a rectangular parallelepiped, the square of any diagonal (AC 1, drawing 82) equal to the sum of the squares of its three dimensions .

Having drawn the diagonal of the base of the AC, we get the triangles AC 1 C and ACB. Both of them are rectangular: the first because the parallelepiped is straight and, therefore, the edge CC 1 is perpendicular to the base; the second because the parallelepiped is rectangular and, therefore, a rectangle lies at its base. From these triangles we find:

AC 1 2 = AC 2 + CC 1 2 and AC 2 = AB 2 + BC 2

Hence,

AC 1 2 = AB 2 + BC 2 + CC 1 2 = AB 2 + AD 2 + AA 1 2.

Consequence.In a rectangular parallelepiped, all diagonals are equal.

It will be useful for senior students to learn how to solve the USE problems to find the volume and other unknown parameters of a rectangular parallelepiped. The experience of previous years confirms the fact that such tasks are quite difficult for many graduates.

At the same time, high school students with any level of training should understand how to find the volume or area of ​​a rectangular parallelepiped. Only in this case will they be able to expect to receive competitive points based on the results of passing the unified state exam in mathematics.

The main nuances to remember

• The parallelograms that make up a parallelepiped are its faces, their sides are edges. The vertices of these figures are considered to be the vertices of the polyhedron itself.
• All diagonals of a rectangular parallelepiped are equal. Since this is a straight polyhedron, the side faces are rectangles.
• Since a parallelepiped is a prism with a parallelogram at its base, this figure has all the properties of a prism.
• The side edges of the rectangular parallelepiped are perpendicular to the base. Therefore, they are his heights.

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The prism is called parallelepiped if its bases are parallelograms. Cm. Fig. 1.

Box properties:

Opposite faces of a parallelepiped are parallel (i.e., lie in parallel planes) and equal.

The diagonals of the parallelepiped intersect at one point and are halved by this point.

Adjacent faces of a parallelepiped- two faces that have a common edge.

Opposite faces of a parallelepiped- faces that do not have common edges.

Opposite vertices of a parallelepiped- two vertices that do not belong to the same face.

Diagonal of a parallelepiped- a segment that connects opposite vertices.

If the side edges are perpendicular to the planes of the bases, then the parallelepiped is called direct.

A straight parallelepiped, the bases of which are rectangles, is called rectangular... A prism, all faces of which are squares, is called cube.

Parallelepiped- a prism whose bases are parallelograms.

Straight parallelepiped- a parallelepiped with side edges perpendicular to the plane of the base.

Rectangular parallelepiped Is a straight parallelepiped, the bases of which are rectangles.

Cube- rectangular parallelepiped with equal edges.

Parallelepiped called a prism, the base of which is a parallelogram; thus, a parallelepiped has six faces, and they are all parallelograms.

Opposite faces are pairwise equal and parallel. The parallelepiped has four diagonals; they all intersect at one point and are divided in half at it. Any face can be taken as a base; the volume is equal to the product of the base area by the height: V = Sh.

A parallelepiped, the four side faces of which are rectangles, is called straight.

A straight parallelepiped, in which all six faces are rectangles, is called rectangular. Cm. Fig. 2.

The volume (V) of a straight parallelepiped is equal to the product of the base area (S) and the height (h): V = Sh .

For a rectangular parallelepiped, in addition, the following formula holds: V = abc, where a, b, c are edges.

The diagonal (d) of a rectangular parallelepiped is related to its edges by the relation d 2 = a 2 + b 2 + c 2 .

Rectangular parallelepiped- a parallelepiped in which the side edges are perpendicular to the bases, and the bases are rectangles.

Properties of a rectangular parallelepiped:

In a rectangular parallelepiped, all six faces are rectangles.

All dihedral corners of a rectangular parallelepiped are straight.

The square of the diagonal of a rectangular parallelepiped is equal to the sum of the squares of its three dimensions (the lengths of three edges having a common vertex).

The diagonals of a rectangular parallelepiped are equal.

A rectangular parallelepiped, all of whose faces are squares, is called a cube. All the edges of the cube are equal; volume (V) of a cube is expressed by the formula V = a 3, where a is the edge of the cube.

In this lesson, everyone will be able to study the topic "Rectangular parallelepiped". At the beginning of the lesson, we will repeat what an arbitrary and straight parallelepiped are, recall the properties of their opposite faces and diagonals of a parallelepiped. Then we will consider what a rectangular parallelepiped is and discuss its main properties.

Topic: Perpendicularity of lines and planes

Lesson: Rectangular Parallelepiped

A surface made up of two equal parallelograms ABCD and A 1 B 1 C 1 D 1 and four parallelograms ABB 1 A 1, BCC 1 B 1, CDD 1 C 1, DAA 1 D 1 is called parallelepiped(fig. 1).

Fig. 1 Parallelepiped

That is: we have two equal parallelograms ABCD and A 1 B 1 C 1 D 1 (base), 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 parallelograms that make up the parallelepiped.

1. Opposite faces of the box are parallel and equal.

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

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 the parallelepiped intersect at one point and are halved 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 by this point in half (Fig. 2).

Fig. 2 The diagonals of the parallelepiped intersect and are halved by the intersection point.

3. There are three quadruples of equal and parallel parallelepiped edges: 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 lateral 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 rectangles lie in the side faces. Arbitrary parallelograms lie at the bases. Denote, ∠BAD = φ, the angle φ can be any.

Fig. 3 Straight parallelepiped

So, a straight 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 ribs are perpendicular to the base. The bases are rectangles.

Parallelepiped ABCDA 1 B 1 C 1 D 1 - 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 °, that is, there is a rectangle at the base.

Fig. 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 rectangular parallelepiped.

So, rectangular parallelepiped is a parallelepiped with side edges perpendicular to the base. The base of the rectangular parallelepiped is a rectangle.

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

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

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

3. All dihedral corners of a rectangular parallelepiped are straight.

Consider, for example, the dihedral angle of a rectangular parallelepiped with an edge AB, that is, the dihedral angle between the planes ABB 1 and ABC.

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

Take point A on edge AB. AA 1 - perpendicular to the AB edge in the ABB-1 plane, AD perpendicular to the AB edge in the ABC plane. Hence, ∠А 1 АD is the linear angle of the given dihedral angle. ∠А 1 АD = 90 °, which means that the dihedral angle at the edge AB is 90 °.

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

It is proved in a similar way that any dihedral angles of a rectangular parallelepiped are straight.

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 outgoing from one vertex of the rectangle are the dimensions of the rectangular parallelepiped. They are sometimes called length, width, height.

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

Prove:.

Fig. 5 Rectangular parallelepiped

Evidence:

Straight line CC 1 is perpendicular to the plane ABC, and hence the straight line AC. This means that triangle CC 1 A is rectangular. By the Pythagorean theorem:

Consider a right-angled triangle ABC. By the Pythagorean theorem:

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

As , but then. Since CC 1 = AA 1, then what was required to prove.

The diagonals of a rectangular parallelepiped are equal.

Let's designate the measurements of the parallelepiped ABC as a, b, c (see Fig. 6), then AC 1 = CA 1 = B 1 D = DB 1 =