What is the base of the parallelepiped. Rectangular parallelepiped. What elements can this figure be divided into

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

Topic: Parallelism of lines and planes

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

In this lesson, we will give a definition of 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 that are in parallel planes. Designation: ABCDА 1 B 1 C 1 D 1 or AD 1 (Fig. 1.).

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

1. Geometry. Grade 10-11: textbook for students educational institutions(basic and profile levels) / I. M. Smirnova, V. A. Smirnov. - 5th edition, corrected and supplemented - M.: Mnemozina, 2008. - 288 p.: ill.

Tasks 10, 11, 12 page 50

2. Construct a section of a rectangular parallelepiped ABCDА1B1C1D1 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 coming out of the same vertex, and calculate its perimeter and area.

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

A parallelepiped is a prism whose bases are parallelograms. In this case, all edges will parallelograms.
Each parallelepiped can be considered as a prism in three different ways, since every two opposite faces can be taken as bases (in Fig. 5, faces ABCD and A "B" C "D", or ABA "B" and CDC "D", or BC "C" and ADA "D").
The body under consideration has twelve edges, four equal and parallel to each other.
Theorem 3 . The diagonals of the parallelepiped intersect at one point, coinciding with the midpoint of each of them.
The parallelepiped ABCDA"B"C"D" (Fig. 5) has four diagonals AC", BD", CA", DB". We must prove that the midpoints of any two of them, for example, AC and BD, coincide. This follows from the fact that the figure ABC "D", which has equal and parallel sides AB and C "D", is a parallelogram.
Definition 7 . A right parallelepiped is a parallelepiped that is also a straight prism, that is, a parallelepiped whose side edges are perpendicular to the base plane.
Definition 8 . A rectangular parallelepiped is a right parallelepiped whose base is a rectangle. In this case, all its faces will be rectangles.
A rectangular parallelepiped is a right prism, no matter which of its faces we take as the base, since each of its edges is perpendicular to the edges coming out of the same vertex with it, and will, therefore, be perpendicular to the planes of the faces defined by these edges. In contrast, a straight, but not rectangular, box can be viewed as a right prism in only one way.
Definition 9 . The lengths of three edges of a cuboid, of which no two are parallel to each other (for example, three edges coming out of the same vertex), are called its dimensions. Two |rectangular parallelepipeds having correspondingly equal dimensions are obviously equal to each other.
Definition 10 A cube is a rectangular parallelepiped, all three dimensions of which are equal to each other, so that all its faces are squares. Two cubes whose edges are equal are equal.
Definition 11 . An inclined parallelepiped in which all edges are equal and the angles of all faces are equal or complementary is called a rhombohedron.
All faces of a rhombohedron are equal rhombuses. (The shape of a rhombohedron is found in some crystals of great importance, such as crystals of Iceland spar.) In a rhombohedron one can find such a vertex (and even two opposite vertices) that all angles adjacent to it are equal to each other.
Theorem 4 . The diagonals of a rectangular parallelepiped are equal to each other. The square of the diagonal is equal to the sum of the squares of three dimensions.
In a rectangular parallelepiped ABCDA "B" C "D" (Fig. 6), the diagonals AC "and BD" are equal, since the quadrilateral ABC "D" is a rectangle (line AB is perpendicular to the plane BC "C", in which BC lies ") .
In addition, AC" 2 =BD" 2 = AB2+AD" 2 based on the hypotenuse square theorem. But based on the same theorem AD" 2 = AA" 2 + +A"D" 2; hence we have:
AC "2 \u003d AB 2 + AA" 2 + A "D" 2 \u003d AB 2 + AA "2 + AD 2.

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

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

For example:

ABCD \u003d A 1 B 1 C 1 D 1 (equal parallelograms by definition),

AA 1 B 1 B \u003d 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 \u003d 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 bisect that 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 the parallelepiped intersect and bisect the intersection point.

3. There are three quadruples of equal and parallel edges of the 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, SS 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 the line AA 1 is perpendicular to the lines AD and AB, which lie in the plane of the base. And, therefore, rectangles lie in the side faces. And the bases are arbitrary parallelograms. Denote, ∠BAD = φ, the angle φ can be any.

Rice. 3 Right box

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

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

The parallelepiped АВСДА 1 В 1 С 1 D 1 is rectangular (Fig. 4) if:

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

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

Rice. 4 Cuboid

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

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

1. In a cuboid, 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 side faces of a cuboid are rectangles.

3. All dihedral angles of a cuboid are right angles.

Consider, for example, the dihedral angle of a rectangular parallelepiped with an edge AB, i.e., the dihedral angle between the planes ABB 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 considered dihedral angle can also be denoted as follows: ∠А 1 АВD.

Take point A on edge AB. AA 1 is perpendicular to the edge AB in the plane ABB-1, AD is perpendicular to the edge AB in the plane ABC. Hence, ∠A 1 AD is the linear angle of the given dihedral angle. ∠A 1 AD \u003d 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 similarly that any dihedral angles of a rectangular parallelepiped are right.

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

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

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

Prove: .

Rice. 5 Cuboid

Proof:

The line CC 1 is perpendicular to the plane ABC, and hence to the line AC. So triangle CC 1 A is a right triangle. According to the Pythagorean theorem:

Consider right triangle ABC. According to the Pythagorean theorem:

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

Because , a , then. Since CC 1 = AA 1, then what was required to be proved.

The diagonals of a rectangular parallelepiped are equal.

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

Theorem. In any parallelepiped, opposite faces are equal and parallel.

So, the faces (Fig.) 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. These faces 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 .

Theorem. In any parallelepiped, all four diagonals intersect at one point and are divided in half at it.

Take (fig.) in a parallelepiped any two diagonals, for example, AC 1 and DB 1, and draw straight lines AB 1 and DC 1.

Since the edges AD and B 1 C 1 are respectively equal and parallel to the edge BC, they are equal and parallel to each other.

As a result, the figure ADC 1 B 1 is a parallelogram in which C 1 A and DB 1 are diagonals, and in the parallelogram the diagonals intersect in half.

This proof can be repeated for every two diagonals.

Therefore, the diagonal AC 1 intersects with BD 1 in half, the diagonal BD 1 with A 1 C in half.

Thus, all diagonals intersect in half and, therefore, at one point.

Theorem. In a cuboid, the square of any diagonal is equal to the sum of the squares of its three dimensions.

Let (fig.) AC 1 be some diagonal of a rectangular parallelepiped.

After drawing AC, we get two triangles: AC 1 C and ACB. Both are rectangular.

the first because the box is straight, and therefore the edge CC 1 is perpendicular to the base,

the second is because the parallelepiped is rectangular, which means that it has a rectangle at its base.

From these triangles we find:

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

Therefore, AC 2 1 = AB 2 + BC 2 + СС 2 1 = AB 2 + AD 2 + AA 2 1

Consequence. In a cuboid, all diagonals are equal.

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