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 a-priory),
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 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 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.
Square diagonal of a cuboid equal to the sum 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:
Let's 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 , 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 =
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 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 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 =
Let's say Achilles runs ten times faster than the tortoise and is a thousand steps behind it. During the time it takes Achilles to run this distance, the tortoise will crawl a hundred steps in the same direction. When Achilles runs a hundred steps, the tortoise crawls another ten steps, and so on. The process will continue ad infinitum, Achilles will never catch up with the tortoise.
This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Hilbert... They all considered Zeno's aporia in one way or another. The shock was so strong that " ...discussions continue to this day, to reach a common opinion about the essence of paradoxes scientific community so far it has not been possible... mathematical analysis, set theory, new physical and philosophical approaches were involved in the study of the issue; none of them became a generally accepted solution to the problem..."[Wikipedia, "Zeno's Aporia". Everyone understands that they are being fooled, but no one understands what the deception consists of.
From a mathematical point of view, Zeno in his aporia clearly demonstrated the transition from quantity to . This transition implies application instead of permanent ones. As far as I understand, the mathematical apparatus for using variable units of measurement has either not yet been developed, or it has not been applied to Zeno’s aporia. Applying our usual logic leads us into a trap. We, due to the inertia of thinking, apply constant units of time to the reciprocal value. From a physical point of view, this looks like time slowing down until it stops completely at the moment when Achilles catches up with the turtle. If time stops, Achilles can no longer outrun the tortoise.
If we turn our usual logic around, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of his path is ten times shorter than the previous one. Accordingly, the time spent on overcoming it is ten times less than the previous one. If we apply the concept of “infinity” in this situation, then it would be correct to say “Achilles will catch up with the turtle infinitely quickly.”
How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal units. In Zeno's language it looks like this:
In the time it takes Achilles to run a thousand steps, the tortoise will crawl a hundred steps in the same direction. During the next time interval equal to the first, Achilles will run another thousand steps, and the tortoise will crawl a hundred steps. Now Achilles is eight hundred steps ahead of the tortoise.
This approach adequately describes reality without any logical paradoxes. But this is not a complete solution to the problem. Einstein’s statement about the irresistibility of the speed of light is very similar to Zeno’s aporia “Achilles and the Tortoise”. We still have to study, rethink and solve this problem. And the solution must be sought not in infinitely large numbers, but in units of measurement.
Another interesting aporia of Zeno tells about a flying arrow:
A flying arrow is motionless, since at every moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.
In this aporia, the logical paradox is overcome very simply - it is enough to clarify that at each moment of time a flying arrow is at rest at different points in space, which, in fact, is motion. Another point needs to be noted here. From one photograph of a car on the road it is impossible to determine either the fact of its movement or the distance to it. To determine whether a car is moving, you need two photographs taken from the same point at different points in time, but you cannot determine the distance from them. To determine the distance to a car, you need two photographs taken from different points in space at one point in time, but from them you cannot determine the fact of movement (of course, you still need additional data for calculations, trigonometry will help you). What I want to point out Special attention, is that two points in time and two points in space are different things that should not be confused, because they provide different opportunities for research.
The differences between set and multiset are described very well on Wikipedia. Let's see.
As you can see, “there cannot be two identical elements in a set,” but if there are identical elements in a set, such a set is called a “multiset.” Reasonable beings will never understand such absurd logic. This is the level of talking parrots and trained monkeys, who have no intelligence from the word “completely”. Mathematicians act as ordinary trainers, preaching to us their absurd ideas.
Once upon a time, the engineers who built the bridge were in a boat under the bridge while testing the bridge. If the bridge collapsed, the mediocre engineer died under the rubble of his creation. If the bridge could withstand the load, the talented engineer built other bridges.
No matter how mathematicians hide behind the phrase “mind me, I’m in the house,” or rather, “mathematics studies abstract concepts,” there is one umbilical cord that inextricably connects them with reality. This umbilical cord is money. Let us apply mathematical set theory to mathematicians themselves.
We studied mathematics very well and now we are sitting at the cash register, giving out salaries. So a mathematician comes to us for his money. We count out the entire amount to him and lay it out on our table in different piles, into which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his “mathematical set of salary.” Let us explain to the mathematician that he will receive the remaining bills only when he proves that a set without identical elements is not equal to a set with identical elements. This is where the fun begins.
First of all, the logic of the deputies will work: “This can be applied to others, but not to me!” Then they will begin to reassure us that bills of the same denomination have different bill numbers, which means they cannot be considered the same elements. Okay, let's count salaries in coins - there are no numbers on the coins. Here the mathematician will begin to frantically remember physics: on different coins there is different quantities mud, crystal structure and the arrangement of atoms in each coin is unique...
And now I have the most interest Ask: where is the line beyond which the elements of a multiset turn into elements of a set and vice versa? Such a line does not exist - everything is decided by shamans, science is not even close to lying here.
Look here. We select football stadiums with the same field area. The areas of the fields are the same - which means we have a multiset. But if we look at the names of these same stadiums, we get many, because the names are different. As you can see, the same set of elements is both a set and a multiset. Which is correct? And here the mathematician-shaman-sharpist pulls out an ace of trumps from his sleeve and begins to tell us either about a set or a multiset. In any case, he will convince us that he is right.
To understand how modern shamans operate with set theory, tying it to reality, it is enough to answer one question: how do the elements of one set differ from the elements of another set? I'll show you, without any "conceivable as not a single whole" or "not conceivable as a single whole."
The sum of the digits of a number is a dance of shamans with a tambourine, which has nothing to do with mathematics. Yes, in mathematics lessons we are taught to find the sum of the digits of a number and use it, but that’s why they are shamans, to teach their descendants their skills and wisdom, otherwise shamans will simply die out.
Do you need proof? Open Wikipedia and try to find the page "Sum of digits of a number." She doesn't exist. There is no formula in mathematics that can be used to find the sum of the digits of any number. After all, numbers are graphic symbols with which we write numbers, and in the language of mathematics the task sounds like this: “Find the sum of graphic symbols representing any number.” Mathematicians cannot solve this problem, but shamans can do it easily.
Let's figure out what and how we do in order to find the sum of the digits of a given number. And so, let us have the number 12345. What needs to be done in order to find the sum of the digits of this number? Let's consider all the steps in order.
1. Write down the number on a piece of paper. What have we done? We have converted the number into a graphical number symbol. This is not a mathematical operation.
2. We cut one resulting picture into several pictures containing individual numbers. Cutting a picture is not a mathematical operation.
3. Convert individual graphic symbols into numbers. This is not a mathematical operation.
4. Add the resulting numbers. Now this is mathematics.
The sum of the digits of the number 12345 is 15. These are the “cutting and sewing courses” taught by shamans that mathematicians use. But that is not all.
From a mathematical point of view, it does not matter in which number system we write a number. So, in different systems In calculus, the sum of the digits of the same number will be different. In mathematics, the number system is indicated as a subscript to the right of the number. WITH a large number 12345 I don’t want to fool my head, let’s look at the number 26 from the article about . Let's write this number in binary, octal, decimal and hexadecimal number systems. We won't look at every step under a microscope; we've already done that. Let's look at the result.
As you can see, in different number systems the sum of the digits of the same number is different. This result has nothing to do with mathematics. It’s the same as if you determined the area of a rectangle in meters and centimeters, you would get completely different results.
Zero looks the same in all number systems and has no sum of digits. This is another argument in favor of the fact that. Question for mathematicians: how is something that is not a number designated in mathematics? What, for mathematicians nothing exists except numbers? I can allow this for shamans, but not for scientists. Reality is not just about numbers.
The result obtained should be considered as proof that number systems are units of measurement for numbers. After all, we cannot compare numbers with different units measurements. If the same actions with different units of measurement of the same quantity lead to different results after comparing them, it means it has nothing to do with mathematics.
What is real mathematics? This is when the result of a mathematical operation does not depend on the size of the number, the unit of measurement used and on who performs this action.
Oh! Isn't this the women's restroom?
- Young woman! This is a laboratory for the study of the indephilic holiness of souls during their ascension to heaven! Halo on top and arrow up. What other toilet?
Female... The halo on top and the arrow down are male.
If such a work of design art flashes before your eyes several times a day,
Then it’s not surprising that you suddenly find a strange icon in your car:
Personally, I make an effort to see minus four degrees in a pooping person (one picture) (a composition of several pictures: a minus sign, the number four, a designation of degrees). And I don’t think this girl is a fool who doesn’t know physics. She just has an arch stereotype of perception graphic images. And mathematicians teach us this all the time. Here's an example.
1A is not “minus four degrees” or “one a”. This is "pooping man" or the number "twenty-six" in hexadecimal notation. Those people who constantly work in this number system automatically perceive a number and a letter as one graphic symbol.
A parallelepiped is a prism whose bases are parallelograms. In this case, all edges will be parallelograms.
Each parallelepiped can be considered as a prism with three different ways, since every two opposite faces can be taken as bases (in Figure 5, faces ABCD and A"B"C"D", or ABA"B" and CDC"D", or VSV"C" and ADA"D") .
The body in question has twelve edges, four equal and parallel to each other.
Theorem 3
. The diagonals of a parallelepiped intersect at one point, coinciding with the middle 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", having 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 plane of the base.
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 emerging from the same vertex, and will, therefore, be perpendicular to the planes of the faces defined by these edges. In contrast, a straight, but not rectangular, parallelepiped can be viewed as a right prism in only one way.
Definition 9
. The lengths of three edges of a rectangular parallelepiped, of which no two are parallel to each other (for example, three edges emerging from 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
. Inclined parallelepiped, in which all edges are equal to each other and the angles of all faces are equal or complementary, is called a rhombohedron.
All faces of a rhombohedron are equal rhombuses. (Some crystals have a rhombohedron shape, having great importance, for example, Iceland spar crystals.) In a rhombohedron you can find a vertex (and even two opposite vertices) such 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 the three dimensions.
In the rectangular parallelepiped ABCDA"B"C"D" (Fig. 6), the diagonals AC" and BD" are equal, since the quadrilateral ABC"D" is a rectangle (the straight line AB is perpendicular to the plane ECB"C", in which BC lies") .
In addition, AC" 2 =BD" 2 = AB2+AD" 2 based on the theorem about the square of the hypotenuse. But based on the same theorem AD" 2 = AA" 2 + +A"D" 2; hence we have:
AC" 2 = AB 2 + AA" 2 + A" D" 2 = AB 2 + AA" 2 + AD 2.
