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Scho also diagonal parallelepipeda. Diagonal parallelepipeda. Formula. How can I know the diagonal parallelepipeda? - Good information for everyone. Taking care of your confidentiality at the same company

At the end of the day, you can learn the theme "Straightforward parallelepiped". On the ear of the lesson, we can repeat it, but also a decent and straight parallel line, guessed by the power of the other sides and diagonals of the parallel line. It is clear that it is also straight-forward parallelepiped, and the main power is discussed.

Topic: Perpendicularity of straight lines and areas

Lesson: Rectangular parallelepiped

The top, folded from two parallelograms ABCD and A 1 B 1 C 1 D 1 and some parallelograms ABB 1 A 1, BCC 1 B 1, CDD 1 C 1, DAA 1 D 1, be called parallelepiped(Fig. 1).

Small. 1 Parallelepiped

Tobto: at most two parallelograms ABCD and A 1 B 1 C 1 D 1 (aside), the stench lie in parallel areas so that the ribs AA 1, BB 1, DD 1, CC 1 are parallel. This rank, folded from parallelograms on the surface, is called parallelepiped.

In such a rank, the surface of the parallelepiped is the sum of all parallelograms, including the number of parallelepipeds.

1. The prototypes are parallel and parallel.

(Figuri rivni, so you can have one overlap)

for example:

ABCD = A 1 B 1 C 1 D 1 (equal parallelograms for values),

АА 1 В 1 В = DD 1 С 1 С (so yak АА 1 В 1 В і DD 1 С 1 С - opposite sides of parallelepiped),

AA 1 D 1 D = BB 1 C 1 C (so yak AA 1 D 1 D і BB 1 C 1 C - opposite sides of the parallelepiped).

2. Diagonals of parallelepipeds are interlaced in one point and spread over a single point of navpil.

Diagonal parallelepipeda AC 1, B 1 D, A 1 C, D 1 B are interlaced in one point O, and the skin diagonal extends over a whole point of navpil (Fig. 2).

Small. 2 Diagonals of parallelepipeds are overturned and overlapped by a point.

3. Є three quadruples of parallel and parallel ribs of 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.

Viznachennya. Parallelepiped is called straight, as well as the ribs perpendicular to the sides.

Do not bichne rib AA 1 perpendicular to the base (Fig. 3). Tse means that the straight line AA 1 is perpendicular to the straight lines AD and AB, which lie in the area of \ u200b \ u200bpart. And, it means, in bichnyh facets lie rectangulars. And in pіdstavah lie pretty parallelograms. Apparently, ∠BAD = φ, qut φ can be-yak.

Small. 3 Straight parallelepiped

Otzhe, straight parallelepiped - tse parallelepiped, in which bichni ribs perpendicular to the paralelepiped.

Viznachennya. Parallelepiped is called straight-headed, every single rib is perpendicular to the base. Pidstavi є rectangular.

Parallelepiped ABCDE 1 B 1 C 1 D 1 - rectangular (Fig. 4), where:

1. AA 1 ⊥ ABCD (sometimes the edge is perpendicular to the area of the front, so that it is parallel to the straight line).

2. ∠BAD = 90 °, ie. In the basis there is a rectangle.

Small. 4 Rectangular parallelepiped

The upright parallelepiped has all the power of the pre-parallelepiped. Alle additional power, as a result of the designation of a rectangular parallelepiped.

Otzhe, rectangular parallelepiped- tse parallelepiped, at which bichni ribs are perpendicular to the base. Pidstava rectangular parallelepipeda - rectangular.

1. The rectangular parallelepiped has all the number of faces of the rectangular.

ABCD і А 1 В 1 С 1 D 1 - rectangulars for design.

2. Bichni ribs perpendicular to the base... This means that all the biches of the rectangular parallelepiped are rectangulars.

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

It is clear, for example, a dihedral cut of a rectangular parallelepiped with an edge AB, i.e., a dihedral cut between areas ABB 1 and ABC.

AB is an edge, point A 1 lies in one area - in area ABB 1, and point D in the other - in area A 1 B 1 C 1 D 1. The appearance of the two-sided cut can be significant by the following rank: ∠A 1 ABD.

Go to point A on the edge AB. AA 1 - perpendicular to edge AB in area ABB-1, AD perpendicular to edge AB in area ABC. This means that ∠А 1 АD is the line cut of the given dihedral cut. ∠А 1 АD = 90 °, which means, a two-sided kut at the edges AB doors 90 °.

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

Similarly, it should be done, whether it be two-sided kuti of a rectangular parallelepiped straight.

Square of diagonal rectangular parallelepiped for transporting squares three times.

Note. Dovzhini three ribs, which go from one vertex of a rectangular parallelepiped, є vimirs of a rectangular parallelepiped. Їх inodі name the amount, width, height.

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

Bring :.

Small. 5 Rectangular parallelepiped

Delivered:

Straight SS 1 is perpendicular to the area ABC, and hence, straight AC. This means that the tricycle SS 1 A is rectangular. According to Pyfagor's theorem:

Rectangular tricycle ABC is clearly visible. According to Pyfagor's theorem:

Ale ВС and AD - opposite sides of the rectangle. Hence, BC = AD. Todi:

So yak , a , That. Oskilki SS 1 = AA 1, then it will be necessary to bring it up.

Diagonals of rectangular parallelepiped pivni.

Apparently, vimіryuvannya parallelepipeda ABC yak a, b, c (div. Fig. 6), toodi AC 1 = CA 1 = B 1 D = D B 1 =

Theorem. Every parallelepiped has opposite edges and parallel edges.

So, between (Fig.) BB 1 C 1 C and AA 1 D 1 D parallel, that is, two intersecting straight lines BB 1 and B 1 З 1 one side parallel to two intersecting straight lines AA 1 and A 1 D 1 small. Ci between and є equal, so yak B 1 З 1 = A 1 D 1, B 1 B = A 1 A (yak opposite sides of the parallelogram) і ∠BB 1 С 1 = ∠AA 1 D 1.

Theorem. In every parallelepiped, all the diagonals are intertwined in one point and spread in different directions.

At the same time (Fig.) In parallel with two diagonals, for example, AC 1 and DB 1, and it is carried out straight AB 1 and DC 1.

So, as the edges AD and B 1 З 1 are similar and parallel to the edge BC, then the smell is equal and parallel to each other.

Occasionally, ADC 1 B 1 є parallelogram, in which Z 1 A and DB 1 - diagonal, and in the parallelogram the diagonal changes into navpil.

This can be proven to be repeated about the skin two diagonals.

To this, the diagonal AC 1 is overflowed from BD 1 navpil, diagonal BD 1 s A 1 S navpil.

In such a rank, all diagonals are intertwined into one point.

Theorem. In a rectangular parallelepiped square, be-diagonal squares in three squares.

Nekhai (Fig.) AC 1 є as the diagonal of the rectangular parallelepiped.

Having carried out AC, two tricycles are accepted: AC 1 С and ACB. The offense of the stench is straightforward:

First of all, it is parallel straight, and the same, the edge CC 1 is perpendicular to the base,

the other is that it is parallel rectangular, which means that it is rectangular in front of it.

3 tsikh trikutnik_v known:

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

From the same, AC 2 +1 = AB 2 + BC 2 + CC 2 +1 = AB 2 + AD 2 + AA 2 1

Slidstvo. In the rectangular parallelepiped all diagonals are.

Students of the senior classes will have a chance to see the knowledge of the EDI's knowledge of the amount of non-native parameters of the rectangular parallelepiped. Above all, there is the fact that it’s possible to carry out folding for bagatech vypusknikov.

With a lot of reason, how to know about the area of the rectangular parallelepiped, the senior pupils are guilty for being a little bit of training. Just because of the stink, you can get rid of competing scores for the baggage of one state college in mathematics.

Basic nuances, like varto zapam'yatati

Parallelograms, which are folded into parallelepipeds, є of edges, їх sides - edges. The tops of the tsikh figur vvazhayutsya tops of the most verge.
All diagonals of the rectangular parallelepiped are equal. So, like a straight sided, the many sides are rectangulars.
So, like a parallelepiped - a prism, in the basis of which there is a parallelogram, a figure of all power is prisms.
Bichni ribs of a rectangular parallelepiped perpendicular to the base. Otzhe, stench є yogo visot.

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