Chapter IV. Straight lines and planes in space. Polyhedra
Problems for Chapter IV
4.1. How many planes in space can be drawn: a) through a point; b) in two various points; c) through three different points that do not lie on the same line; d) through three different points; d) through four points?
4.2. How many planes in space can be drawn: a) through one straight line; b) through two intersecting lines; c) through two arbitrary lines?
4.3. How many planes in space can be drawn: a) through a straight line and a point; b) through two intersecting lines and a point?
4.4. There are four points in space, no three of them belong to the same line. A straight line is drawn through each pair of given points. How many of these lines can be held?
4.5. There are four points in space, no three of them belong to the same line. Through every three of these points a plane is drawn. How many such planes can be drawn?
4.6. Is the statement true: if a straight line l 1 intersects the line l 2 and straight l 2 intersects the line l 3, then straight l 1 intersects the line l 3 ?
4.7. Is the statement true: if straight l 1 , l 2 crossed and straight l 2 , l 3 crossing then l 1 and l 3 interbreeding?
4.8. How many pairs of crossing edges, i.e., edges lying on crossing lines, are there in a triangular pyramid?
4.9. How many pairs of parallel and crossing edges are there in a parallelepiped?
4.10. Prove that there is only one plane passing through two parallel lines.
4.11. How to construct a straight line that intersects:
a) with each of two intersecting lines;
b) with each of two parallel lines?
4.12. How many planes are parallel to a line? l, can we draw through a given point A outside this line?
4.13. Straight l parallel to the plane R. How many lines are parallel to the line l, can be drawn in the plane R? What's it like mutual arrangement all these straight lines?
4.14. It is known that straight l parallel to the line T, which is parallel to the plane R. Will there be a direct l parallel to the plane R?
4.15. Let straight l And T parallel, and one plane is drawn through each of them. Prove that if these planes intersect, then the line of their intersection is parallel to the lines l And T.
4.16. Prove that if a plane intersects one of two parallel lines, then it also intersects the other.
4.17. Prove that if a line intersects one of the parallel planes, then it also intersects the other.
4.18. Prove that if the plane R 1 parallel to the plane R 2, a R 2 parallel to the plane R 3 then R 1 parallel R 3. (Property of transitivity.)
4.19. Prove that the segments of parallel lines contained between parallel planes, have equal lengths.
4.20. Construct a plane passing through this line l, parallel to the line T(straight l And T interbreeding).
4.21. Given a cube ABCDA 1 B 1 C 1 D 1 . Find the angle between the lines: a) AD and BB 1 b) AD and A 1 D 1) c) AC and B 1 D 1 d) AC and A 1 D 1.
4.22. Prove that if two lines are perpendicular to one plane, then these lines are parallel.
4.23. Prove that if two planes are perpendicular to one line, then these planes are parallel.
4.24. Segments AB and BC are sides of square ABCD. Planes are drawn through straight lines AB and BC, respectively. R 1 and R 2. Straight l- line of intersection of planes R 1 and R 2, and l _|_ (AB). Prove that (AB) _|_ R 2 .
4.25. Point O is the center of a square with a side T. The segment OM is perpendicular to the plane of the square, |OM| = m / 2. Find the distance from point M to the top of the square.
4.26. Find the distance from point M to the plane equilateral triangle, if the side of this triangle is 3 √ 3 cm, and the distance from the point to each of the vertices of the triangle is 5 cm.
4.27. Find the set of all points in space equidistant from three given points.
4.28. In an isosceles right triangle ABC the legs are equal A see from the top right angle C drawn to the plane /\
ABC is perpendicular to CD, and
| CD | = 2 A cm. Find the distance from point D to the hypotenuse AB.
4.29. Legs right triangle ABC are equal to 4 cm and 3 cm. A perpendicular is drawn through the vertex of right angle C of the triangle P to plane ABC. Find the distance from point M n to the hypotenuse of the triangle, if | MS | = 2.6 cm.
4.30. If the faces of one dihedral angle serve as a continuation of the faces of another, then such dihedral angles are called vertical. Prove that vertical dihedral angles are congruent.
4.31. From point M of the circle, a perpendicular MA is drawn to the plane of the circle bounded by this circle. The diameter MB is drawn from point M; [ВС] - arbitrary chord. Point A is connected to points B and C. Determine the type triangle ABC.
4.32. Prove that if the planes R And q perpendicular and straight 1 R perpendicular to a straight line T = pq, That 1 _|_ q.
4.33. Let three pairwise intersecting planes be given p, q, r. Prove that if
r _|_ r And q _|_ r, then straight T = pq perpendicular to the plane r.
4.34. Prove that if a plane is perpendicular to one of two parallel planes, then it is also perpendicular to the other plane.
4.35. A half-plane having as its edge an edge of a dihedral angle and dividing it into two congruent parts is called bisector. Prove that bisector half-planes of two adjacent corners perpendicular to each other.
4.36. On the model of the cube ABCDA 1 B 1 C 1 D 1, indicate the projections of the following figures onto the plane of the face AA 1 B 1 B: , , , , /\ From 1 NE, /\ ACD, square BB 1 C 1 C.
4.37. Given a cube ABCDA 1 B 1 C 1 D 1 . a) Find the projection of point M on the plane of faces ABCD, AA 1 D 1 D, AA 1 B 1 B. b) Find the projection of point N = [СD 1 ] on the plane of the indicated faces.
4.38. What are the projections of two lines l 1 and l 2 per plane R, If:
a) straight l 1 and l 2 intersect;
b) straight l 1 and l 2 are crossed;
c) straight l 1 and l 2 are parallel. Consider all possible cases.
4.39. Points A and B belong to the plane R; congruent segments AA 1 and BB 1 are perpendicular to the plane R and are located along different sides from her. Find the angles of the quadrilateral AA 1 BB 1 if |AA 1 | = |AB|.
4.40. The hypotenuse of a right triangle is equal to T, its magnitude acute angle 60°. Find the area of the projection of this triangle onto a plane that makes an angle of 30° with the plane of the triangle.
4.41. The sides of the triangle are 3.9 cm, 4.1 cm and 2.8 cm. Find the area of its projection onto a plane that makes an angle of 60° with the plane of the triangle.
4.42. Construct a section of the cube ABCDA 1 B 1 C 1 D 1 with a plane passing through points M, N and K, if
M = A 1, | ND 1 | = | ND |, | DK | == 2| KS |, N, K.
4.43. Construct a section of a cube ABCDA"B"C"D" with an edge A plane passing through the midpoints of the ribs and [B "C"] and the vertices A" and C. Find the cross-sectional area.
4.44. Construct a section of the cube with a plane so that it is a regular hexagon.
4.45. In the tetrahedron MABC, draw sections through the middle of the edge [AB] parallel to the edges: a) [AC] and ; b) [VS] and [SM]; c) [BC] and [AM].
4.46. Find the area of the section drawn through the midpoints of two adjacent lateral edges of a regular quadrangular pyramid with side A and height h perpendicular to the base of the pyramid.
4.47. Is there a trihedral angle whose plane angles are equal to: a) 120°, 97°, 33°;
b) 120°, 120°, 130°; c) 108°, 92°, 160°; d) 157°, 82°, 64°.
4.48. IN trihedral angle two plane angles of 45°, and dihedral angle between them - 90°. Find the third plane angle.
4.49. Base sides right parallelepiped equal 3√2 cm and 14 cm, the angle between them is 135°, side rib 12 cm. Find the diagonals of the parallelepiped.
4.50. Diagonal correct quadrangular prism equal to 9 cm; the total surface of the prism is 144 cm 2. Find the side of the base and the side edge of the prism.
4.51. Full surface rectangular parallelepiped equal to 352 cm 2. Find its measurements if they are in a ratio like 1:2:3.
4.52. The edge of the cube is equal to A. Find the cross-sectional area of the cube by a plane passing through the ends of the edges emerging from one vertex.
4.53. The edge of the cube is equal to A. Find the length of the segment connecting the midpoints of two crossing edges.
4.54. In a regular quadrangular pyramid MABCD, the side of the base is A, the apothem of the pyramid is 3/2 A. Find the height of the pyramid.
4.55. Find the height of a regular quadrangular pyramid if its side edge is equal to T, and the plane angle at the vertex is β.
4.56. Given a pyramid, the height of which is 16 m, and the base area is 512 m 2. Find the cross-sectional area of the pyramid by a plane drawn parallel to the base at a distance of 5 m from the top.
4.57. Find the side edge of a regular quadrangular pyramid whose side of the base is 14 cm and whose area is diagonal section 14 cm 2.
4.58. A rhombus with diagonals of 12 cm and 16 cm serves as the base of the pyramid. The height of the pyramid passes through the point of intersection of the diagonals and is equal to 6.4 cm. Find full surface pyramids.
4.59. The height of a regular quadrangular pyramid is 28 cm, and the side edge
36 cm. Find the side of the base.
4.60. Prove that the side edge is regular triangular pyramid perpendicular to the opposite edge of the base.
4.61. Prove that side surface regular pyramid equal to the area of the base divided by the cosine of the angle between the plane of the side face and the plane of the base.
4.62 Two regular polyhedra have equal edges, and the surface areas are in the ratio √3 : 6. Determine these polyhedra.
4.63. If we designate an edge regular polyhedron through A, then its surface area is S = 5 a 2 √3. Define a polyhedron.
4.64. Find the dihedral angle between the faces of a regular tetrahedron.
4.65. Find the dihedral angle between adjacent faces of a regular octahedron.
4.66. Points M, A, B and C do not belong to the same plane; (MA) _|_ (BC),
(MB) _|_ (AC). Prove that (MC) _|_ (AB).
4.67. Forces act on point A F 1 , F 2 , F 3, and | F 1 ] = 3 N, | F 2 | = 4 N and | F 3 | = 5 N. The magnitude of the angle between the forces F 1 and F 2 is equal to 60°, and the force F 3 is perpendicular to each of them. Find the magnitude of the resultant.
Geometry lesson in 10th grade.
Subject: Parallelism of lines and planes
Target: Systematize students’ knowledge on the topic “Parallelism of lines and planes”, deepen and consolidate students’ knowledge when solving problems, develop students’ spatial understanding
Equipment: computers (program " Open mathematics. Stereometry."), multimedia board, test compiled using a test shell.
During the classes
I Announcement of the topic and purpose of the lesson.
Motivation for learning activities.
WITH Today we are conducting a geometry lesson on the topic “Parallelism of lines and planes” using computer technology. The use of computers expands the possibilities of learning, in particular, stereometry, as it contributes to the development of students’ spatial concepts, helps a clearer formation of geometric concepts, and expands the existing stock of geometric images.
In previous lessons, we examined the main issues of the topic: parallelism of lines in space, parallelism of a line and a plane, parallelism of planes. Let's repeat these questions.
II Updating of basic knowledge.
What lines in space are called parallel? (...lie in the same plane and do not intersect.)
Of interest are direct ones that do not have common points and not parallel. These are?...crossing straight lines. Define skew lines. (...lines that do not intersect and do not lie in the same plane.)
Formulate a sign of parallelism of lines. (Two lines parallel to a third line are parallel.)
In what case are a straight line and a plane called parallel? (...if they don't intersect.)
Formulate a sign of parallelism between a line and a plane. (If straight, not belonging to the plane, is parallel to some line in this plane, then it is parallel to the plane itself.)
When are two planes called parallel? (...if they don't intersect.)
Formulate a sign of parallelism of planes. (If two intersecting lines of one plane are respectively parallel to two lines of another plane, then such planes are parallel.)
III Working with computers.
Let's take a look theoretical material in the program “Open Mathematics. Stereometry." (Program path: D\VCD\Stereometry)
Students review the theory given in Chapter 2: Parallelism in Space
(2.1 Parallelism of lines
2.2 Parallelism of a straight line and a plane
2.2 Parallelism of two planes)
While working with the program, students encounter new concepts such as lemma, test of crossing lines, trace theorem, etc.
IV Work in groups.
One student remains at each computer and works with the test program. (On the desktop there is a shortcut test-w, Test 10th grade, Open.) The test checks and evaluates students' knowledge on the topic of the lesson. Test assignments are attached.
The rest of the students sit at the tables and perform oral solutions to the following problems:
How many cases are there of the relative position of two different lines in space? (Three)
Is it true. Are two lines skewed if there is no plane in which both lines lie? (Yes)
How many pairs of crossing edges does a triangular pyramid have? (Three)
How many pairs of crossing edges does quadrangular pyramid? (Eight)
Given a line a and a point A outside it. How many lines intersecting a can be drawn through point A? (Infinitely many)
Given an alpha plane and point A outside it. How many lines parallel to the alpha plane can be drawn through point A? (Infinitely many)
The group work is over. Test results are viewed. The guys return to the computers and work on the mistakes that were made when working with tests.
V Problem solving.
Working with the Open Mathematics program. Stereometry."
Button: Problems with solutions.
Given intersecting lines a, b and a point T. Draw a line through point T intersecting lines a and b.
In planimetry, the following theorem is true: two lines perpendicular to a third are parallel. Is this theorem valid in stereometry? (No)
Students solve problems collectively, view the solution to problems on the computer, work with the drawing: remove the fill and restore it, rotate the drawing in various directions, increase it and decrease it, etc. Working with a cube model. Find pairs of intersecting, parallel, crossing lines; intersecting and parallel planes, etc.
Button: Tasks.
Students solve problems independently, enter the answer, and analyze its correctness.
VI Summary.
We repeated, systematized, deepened knowledge on the topic of the lesson. We paid attention to problems with crossing lines. Computer program helped to visualize the combinations geometric shapes in space.
Student assessment.
VII Homemade exercise:
Write down the solutions to the problems in your notebook.
Application
Test tasks
none
only one
infinitely many
none or one
only one
infinitely many
one or infinitely many
none or one
none, one or infinitely many
one or infinitely many
none
infinitely many
none or infinitely many
only one
infinitely many
none or one
one or infinitely many
none
only one
infinitely many
none
none or one
only one
one or infinitely many
none
only one
none or one
none or infinitely many
infinitely many
Given two skew lines a and b. How many planes are there passing through a and parallel to b?
How many planes are there that pass through these three different points in space?
In space, a line a and a point M outside a are given. How many planes are there passing through M and parallel to the line a?
Given a plane alpha and a straight line a not lying in it. How many planes are there passing through a and parallel to alpha?
In space, a line a and a point M are given. How many lines are there passing through M and parallel to line a?
Given an alpha plane and a point M outside alpha. How many planes are there passing through M and parallel to the alpha plane?
Note. Test tasks and answers to them are selected randomly. The test can be limited in time.
Lesson objectives:
educational:
- consider possible cases of relative positions of lines in space;
- develop the skill of reading and drawing drawings, spatial configurations, spatial figures to tasks.
developing:
- develop students’ spatial imagination when solving geometric problems, geometric thinking, interest in the subject, cognitive and creative activity students, math speech, memory, attention;
- develop independence in mastering new knowledge.
educational:
- to instill in students a responsible attitude towards educational work, strong-willed qualities;
- to form an emotional culture and a culture of communication,
- develop a sense of patriotism and love for your hometown.
Teaching methods:
- verbal,
- visual,
- active
Forms of training:
Teaching aids: (including technical teaching aids)
- computer,
- multimedia projector,
- screen,
- Printer,
- printed media ( Handout),
- crossword.
Teacher's opening speech.
Using the knowledge we have learned from the planimetry course about the relative position of lines on a plane, we will try to solve the question of the relative position of lines in space.
The lesson was helped to prepare by students Skotnikova Olga and Stefan Yulia, who, using an independent search for photographs of the sights of the city of Khabarovsk, examined various options relative positions of lines in space.
They were not only able to consider various options for the relative position of lines in space, but also performed creative work- created a multimedia presentation.
Presentations creative reports with a brief explanation and historical background of the sights of our city:
For the 150th anniversary of our city, the masters of light did their best and staged a magnificent laser show on the embankment. Slide No. 2
The attention of numerous guests of Khabarovsk is attracted by the monumental monument erected on Komsomolskaya Square. The twenty-two-meter monument perpetuated the memory of heroic feat Far Eastern Red Guards and partisans, who forever liberated the region from the White Guards and foreign invaders. The monument was opened in October 1956. Slide No. 3
The Khabarovsk railway station was built in 1929 and in those years was considered one of the largest and beautiful train stations Far East. Currently, the station has been reconstructed, its interior has been completely changed and it has again acquired the appearance of a Russian station of the 20th century. Slide No. 4
Conclusion based on slides No. 3 No. 4. Slide No. 5
The Khabarovsk airport has international status, is equipped with modern equipment, and the aviation technical base is capable of servicing any type of aircraft, up to the Boeing 747.
A wide network of regular routes connects Khabarovsk with dozens of cities in Russia, the CIS, and far abroad. Comfortable aircraft depart from Khabarovsk airports and return back at the most convenient time for passengers.
Must be taken right decisions for a limited time when controlling aircraft flights depending on their relative position in airspace and at the airport. Slide No. 6
The cliff - this wonderful place has become one of the symbols of Khabarovsk. We can say that the history of the city began from this place.
In 1858 Captain Y.V. Dyachenko landed here with his detachment and decided to establish his camp here. Later it became a military settlement, then the village of Khabarovsk, and now it is the beautiful city of Khabarovsk.
The building has a large balcony, which is a magnificent observation deck, allowing you to see the embankment, the beach and the expanses of the Amur River stretching beyond the horizon. Slide No. 7
Summing up the presentations.
How do you evaluate the creative preparation of your classmates for the lesson?
Let's draw a conclusion. What options for the relative arrangement of lines in space did we learn today in class? Slide No. 8
Consolidation.
Mathematical dictation, students perform on separate sheets of ready-made drawings and are submitted for inspection to assistant consultants, who check and the results of the inspection are entered into a special sheet.
ABCDA 1 B 1 C 1 D 1 - CUB.
K, M, N - MIDDLES OF RIBS
B 1 C 1, D 1 D, D 1 C 1 RESPECTIVELY,
P - POINT OF INTERSECTION
DIAGONAL FACES AA 1 B 1 B.
Determine the relative position of the lines. Slide No. 9,10,11,12,13,14
Self-test. Slide No. 15
SABC - TETRAHEDRON.
K, M, N, P - MIDDLES OF RIBS
SA, SC, AB, BC RESPECTIVELY.
Slide No. 16, 1, 18, 19, 20
Self-test. Slide No. 21
After completing the mathematical dictation - a brief oral explanation with justification for all tasks.
The test is carried out by students according to the handouts and is also submitted for testing to assistant consultants, who check and the test results are entered into a special sheet
How many cases are there of the relative position of two different lines in space?
The text provides a definition of skew lines. Is the following definition correct: “Two lines are said to intersect if there is no plane in which both of these lines lie.”
c) it is impossible to answer unambiguously
How many pairs of crossing edges does a triangular pyramid have?
How many pairs of crossing edges does a quadrangular pyramid have?
Given a line a and a point A outside it. How many lines intersecting a can be drawn through point A?
b) many
In order for two straight lines not to intersect (it is necessary or sufficient) that they intersect.
In order for two lines to be parallel (it is necessary or sufficient) that they lie in the same plane.
Independent work on options
1 option
Given intersecting lines a, b and a point T. Draw a line through point T intersecting lines a and b.
Option 2
Lines a and b are crossed. Draw a line intersecting b and parallel to line a.
Record sheet for the results of mathematical dictation and testing
| Full name | Mathematical dictation | Test | Sm/r | ||||||||||||||
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | ||
Prepare a creative report on the relative position of lines and planes in space.
Summarizing.
Crossword. Slide No. 22,23
FOR YOUTH AND SCHOOLCHILDREN “STEP INTO THE FUTURE”
CHELYABINSK HEAD COORDINATION CENTER
"INTELLECTUALS OF THE XXI CENTURY"
PYRAMID AND CROSSING LINES
Creative work at the X VII Chelyabinsk
urban scientific and practical conference young
researchers and intellectuals "Step into the future"
(section 3.1)
Chelyabinsk, lyceum No. 000, class 10.
Scientific adviser:
mathematic teacher,
Lyceum No. 000.
Chelyabinsk - 2009
Introduction
The greatest and most mysterious of the seven wonders ancient world is the Giza pyramid complex in Egypt, the most impressive of which is the Pyramid of Cheops. Scientists and theologians have been studying the Great Pyramid for many centuries, marveling at the greatness of the gigantic work of its creation. The pyramid was built between 10490 And 10390 years BC. The Cheops pyramid is spoken of as the most perfect structure in the world - the standard of weights and measures. About what's in her geometric shape information about the structure of the Universe is encoded, solar system and man.
The word pyramid comes from the Greek "pyramis" etymologically related to "feast" - "fire", denoting a symbolic representation of the One Divine Flame, the life of all creatures. Initiates of the past considered the pyramid to be the ideal symbol of the Secret Doctrine. Square base pyramid stands for Z earth, its four sides are the four elements of matter or substance, from the combination of which material nature is created. Triangular sides oriented towards the four cardinal directions, symbolizing the opposites of heat and cold (south and north), light and darkness (east and west). The three main chambers of the pyramid correspond to the brain, heart and reproductive system of man, as well as to his three main energy centers. Main purpose Great Pyramid was carefully hidden.
It turned out that the energy of the pyramid shape “can do” a lot: instant coffee, after standing over the pyramid, acquires a natural taste; cheap wines significantly improve their taste; water acquires properties to promote healing, tones the body, reduces the inflammatory reaction after bites, burns and acts as a natural aid to improve digestion; meat, fish, eggs, vegetables, fruits are mummified, but do not spoil; milk does not sour for a long time; cheese doesn't mold...
Is the pyramid so universal? Let's try to use this wonderful figure to solve school problems.
We set the task to find conditions under which it is easy to determine the distance between crossing lines.
Target work– find a method by which you can measure the distance between crossing lines and check this method to solve practical problems.
Object of study in this work are intersecting straight lines.
Research method– constructing a model that helps determine the location of intersecting lines in space.
Method defines subject of study: Relationship between stereometric objects.
During the study, conditions were found under which the problem was solved in a rational way, and an algorithm for applying the pyramid method to solve specific problems was formulated. In the process of work studied existing methods on this topic, and also designed a convenient and rational way solutions to this problem. Basic Concepts
1.1 Crossing lines
During stereometry lessons in the tenth grade, we became acquainted with crossing lines.
In the same textbook we read about the distance between parallel planes and in paragraph 3 about the distance between intersecting straight lines.
Using these materials, we began to solve practical problems. The solutions to the problems were cumbersome and difficult to see in the drawings. That's why this topic I decided to look it up in reference books and other manuals.
1.2 Methods for determining the distances between crossing lines
The magazine “Mathematics for Schoolchildren” this year (No. 1, 2008) published an article “On distance in general and the distance between crossing lines in particular,” which describes in detail everything known methods constructing a common perpendicular to two intersecting lines. Are being considered specific tasks. In the scientific, theoretical and methodological “Mathematics at School” (No. 1, 2008) an article was published “On some methods for calculating the distance between crossing lines.”
It is worth noting that the task of constructing a common perpendicular to two skew lines requires a lot of painstaking work. At the same time, when finding the distance between intersecting lines, there is no need to construct their common perpendicular! Often it is enough just to see (draw) a more suitable segment, the length of which will be the required distance. In this case, it is advisable to rely on one of the following statements.
1. The distance between crossing lines is equal to the distance between parallel planes passing through these lines.
2. The distance between intersecting lines is equal to the distance from one of them to a plane parallel to it passing through the second line.
3. Distance 1 between intersecting lines containing segments AB and CB, respectively, can be calculated using the formula 
where is the angle between straight lines AB and CD, and is the volume of the triangular pyramid ABCD (Fig. 1)
Approaches based on the application of the first two statements, being purely geometric, require the decider to have good spatial imagination. However, the second approach is sometimes more advantageous to implement in coordinate-vector form. IN reference books meets general equation plane - 
in a rectangular coordinate system, then you can apply the one known in the course analytical geometry formula for distance from point M() to the plane defined by this equation:
After studying the material, I began to construct the object under study, using stereometric models available in the mathematics classroom.
As a result, I found a rational way to solve the problem.
2. Theoretical part.
The method I have developed for finding the distance and angle between intersecting lines, which is conventionally called the “Pyramid Method,” makes it possible to solve the problem quickly and rationally.
Why the “pyramid method”? The fact is that when solving problems using this method, a pyramid PABCD is built, and the meaning of such a construction is the statement: “The distance between intersecting lines is equal to the distance from the point, which is the projection of one of the two given intersecting lines onto a plane perpendicular to it, to orthogonal projection another straight line to the same plane.”
In the magazine “Mathematics at School” (No. 6, 1986) he used the above statement and gave examples of solving problems, but the method of construction differs from the “pyramid method”. The entire construction sequence consists of five steps:
1. Let a straight line and intersecting and arbitrary point P belongs to the line.
2. Draw a perpendicular RA to the straight line. Let RA belong to the plane.
3. Let us draw a perpendicular MN from point M, which belongs to the line, to the plane. Let the line PN, which belongs to the plane, intersect the line at point B. Let us draw perpendiculars BC and AD to the plane so that BC = AD, and points C and D belong to the same half-plane and point C belongs to the line. After this, it can be argued that the quadrilateral ABCD is a rectangle, and therefore parallel (PCD) based on the parallelism of the line and the plane.
4. The problem was reduced to finding the distance from the straight line to the PCD plane parallel to it. A line is perpendicular to (PAD) based on the perpendicularity of the line and the plane; planes (ABC) and (PAD) are perpendicular based on the perpendicularity of the planes. Line CD is perpendicular (PAD) since lines CD are also parallel. The planes (PAD) and (PCD) are perpendicular based on the perpendicularity of the planes. Let us draw a perpendicular AK to the line PD of intersection perpendicular planes PAD and PCD. This means that AK will be perpendicular to the plane (ROS). So, segment AK, which is the altitude of the right triangle PAD equal to the distance between intersecting lines and .
5. Drawing KL, point L belongs to the line and LF KA, point F belongs to the line, we obtain that LE is a common perpendicular to two skew lines and . If the intersecting lines intersect at a right angle (coincides with PD or PD belongs to), then the task is significantly simplified, which is often found in many exercises. By the way, not all tasks require taking point M. Above specified method quite simple, but with the help of this approach almost all problems of finding the distance between crossing lines and constructing a common perpendicular to them are instantly solved. The angle between intersecting lines and can be found as angle PCD from the right triangle PDC.
1. Practical part. Building a pyramid. Calculating the distance between intersecting lines
3.1 Task 1. Each edge of a regular triangular prism is equal to A. Determine the distance between the side of the base and the diagonal of the side face that intersects with it.
Solution.
РВSPCS - correct triangular prism. Let's find the distance between BS and RS. We will carry out:
b) AD BC, AD= BC, point A BC.
c) AK PD; TO . From what was previously proven, the segment AK will be equal to the required distance. Applying the area method to the right triangle PAD, we obtain:
AK= AR *AD:PD = A .
3.2.Task 2. The edge of a regular tetrahedron is A. Find the distance between two edges of a tetrahedron that intersect.
CPQR - regular tetrahedron. CO - height of the tetrahedron. We will look for the distance between PC and RQ.
Let's carry out RA RQ. Point A RQ. Since the skew lines PC and RQ intersect under the straight cut (following the theorem of three perpendiculars), the problem is simplified (coincides with PD)). AK is the height of the right triangle PAD and will be the required distance, but of course it is easier to find AK as the height isosceles triangle RAS (AS=AR)
3.3. Problem 3. The edge of the cube is equal to a. Find the shortest distance between the diagonal of the cube and the diagonal of the base of the cube that intersects it.
Solution: - cube. We will look for the distance between PM and RQ. According to the previously proven statement, the segment AK, which is the height of the right triangle PAD, will be equal to the required distance:
3.4. Task 4. Find the distance between the crossing diagonals of adjacent faces of the cube.