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4CREATIVE APPROACHES TO THE SOLUTION OF TASKS IN
SCHOOL TEXTBOOKS
1Z.Q.Erkaboyeva, 2M.G.Teshaeva
1Teacher of mathematics for 34th specialized school of Kushrabot district of Samarkand region 2 Student of mathematics and informatics of Uzbekistan-Finland Pedagogical Institute
https://doi.org/10.5281/zenodo.7682758
Abstract. In this thesis, the concept of the distance between two intersecting straight lines, which is introduced as a problematic issue in general education school textbooks and has wide applications in practice, is described and the methods of finding this distance are described in detail.
Keywords: intersecting straight lines, parallel straight lines, parallel planes, distance between parallel straight lines, distance between parallel planes, volume of the pyramid.
In modern mathematics textbooks, great emphasis is placed on dividing straight lines and finding the distance between them. The reason for this can be shown by the fact that finding the distance between straight lines is widely encountered in the problems of mechanics and differential geometry and has wide applications in life.
Below are some basic definitions of bisecting straight lines and finding the distance between them, with plenty of examples. First, let's define straight lines:
Definition 1. The intersection of two straight lines that do not lie in the same plane in space are called straight lines (Fig. 1).
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Figure 1
Before introducing the concept of distance between two intersecting straight lines, we define the concept of distance between two shapes.
Definition 2. For points F1 and F2 and arbitrary A1 e F1, A2 e F2of the given forms B1 e F1, B2 e F2 if inequality is appropriate. Points < IB1B2I are called the closest points of
the given shapes. The distance between A1 va A2 forms means the distance between their nearest points (if it exists).
Now we present several definitions that are convenient for finding the distance between two intersecting straight lines in practice:
Definition 3. The distance between the nearest points of two intersecting straight lines is called the distance between the straight lines.
Definition 4. The distance between parallel planes where two intersecting straight lines are located is called the distance between straight lines.
Definition 5. The distance from one of the two intersecting straight lines to the parallel plane on which the second intersecting straight line lies is called the distance between these lines. Below we show the process of solving the problem in several ways: Problem solving. A right prism ABCDA1B1C1D1whose base is a square is given. If the side of the base of the prism is 4 and the height is 2V2, find the distance between the diagonals DA1 and CD1 (Fig. 2).
Figure 2
Note: Below (ABC) - the inscription in the form ABC indicates the plane.
1-way. It is known that CDx lies on the diagonal (CB-^D-^). Also, from the fact that DA ^ICBfollows that the straight line DA1 {CB^^D^} is parallel to the plane. So, the distance between the planes DA1 and (CB-^D-^) is equal to the distance between the diagonals DAx and CD1 according to definition 5. In particular, to find this distance, it is enough to find the distance from the point A_1 of the straight line DA1 (CB-^D-^) to the plane (Fig. 3).
Figure 3
We designate the points of intersection of the diagonals of the bases of the given right prism by 0 and 0-^, respectively.
It is known that the planes (ACC-^) and (CB-^D-^Jintersect in the straight line OxC. So, to find the distance from ^point (CB-^D-^) to the plane, it is enough to find the distance Ax from point O-^C to the straight line A-H (Fig. 4).
Figure 4
According to the condition of the problem, C1C — 2^2, C1D1 — 4. So, AC01C1 for C101 — CiC — right triangle. From this comes the equality ^C01C1 — 450 A0. Also, co1c1 — ^ H01A1
AA1H01- in a right triangle, A101 — 2^2 va sin ^H01A1 — Therefore, HA1 — A101 •
sin^H01A1 — 2^2^ — 2.
2-way. (BDA1) and (CB1D1) planes are parallel to each other, because DA1\lCB1,CD1llBA1. Also, the straight lines A1D and CD1 lie on (BDA1) and (CB1D1) planes, respectively. So, to find the distance between these straight lines, it is enough to find the distance between(5D^1) and (CB1D1) planes. In turn, to find the distance between these planes, it is necessary to find the distance from the point O on the plane (BDA1) to the plane (CB1D1).
Figure 5
(0C01) and (CB1D1) planes are perpendicular to each other and intersect in a straight line C01. We pass OH perpendicular from point O to straight line C01. The length of this section OH is equal to the distance from the point O (CB1D1) to the plane, so it is also equal to the distance between the straight lines A1D and CD1 (Fig. 5). We find the length of this section. It is known that AA1 — 001 — 2^2, AO — CO — 2^2. . Also from AOAA1: A10 — C01 — ■JAA1 + AO2 — 4.
OH
A0H01~AC001. Because the 001 - side is general and ^C010 — ^001H. So, — — —. From this, OH —-—-— 2.
C01 ' C01 4
3-way. Method of volumes
It is known that DA111CB1 and CD11IBA1 relations are relevant. From these conditions, the relation (BDA1)H(CB1D1) is derived (Fig. 6).
Figure 6
BCDA1 — Let's look at the pyramid (Fig. 7).
Figure 7
We denote by h the height dropped from the end of this pyramid С to the base BDAx The length of this height is equal to the distance between the diagonals DAx and CD-^. Because this length is equal to the distance from the end C of the diagonal CDx to the plane BDAx lying on the diagonal DA1. We find the length of this height.
From the conditions of the problem, it can be directly found that BD = AC = 4^2, AO = 2^2. From right angle AOAA1 Ог: A10 = C01 = JAA- + AO2 = 4. .
We find the volume of the pyramid BCDA1 — - through the base BDAand h- height:
_ 1 _1 A1O^BD _ qV2
VBCDA1 = ^ = 3 2 ^ = ~3~ • ^.
We find the volume of the pyramid BCDA1-by the base BCD — and the height AA1:
_1 _1 BC •CD _ 1642
VBCDA1 = • = 3 2 = з .
So, VBCDAi = 842^h = 1642 ^h = 2. REFERENCES
1. Mirzaahmedov M.A., Ismailov Sh.N., Amanov A.Q. Geometry. Part II. LLC "Extremum Press", 2017, Tashkent.
2. Pogorolev A.V., Geometry. 3rd edition. "Teacher" creative publishing house. Tashkent-20054.
3. Гусятников П.Б., Резниченко С.В. Векторная алгебра в примерах и задачах. — М.: Высшая школа, 1985. — 232 с. Архивная копия от 10 января 2014 на Wayback Machine. https://shkolkovo.net/
