Repetition of the theory and solution of typical problems on the perpendicularity of a line and a plane (continued). Perpendicular line and plane, sign and conditions of perpendicularity of line and plane Lesson: Repetition of theory and solution of typical problems on

In this lesson we will review the theory we have covered and continue solving typical problems on the perpendicularity of a line and a plane.
First, let us repeat the theorem-test of the perpendicularity of a line and a plane. And further we will solve problems using this feature.

Topic: Perpendicularity of lines and planes

Lesson: Repetition of theory and solving typical problems on

perpendicularity of a line and a plane (continued)

In this lesson we will review the theory we have covered and continue solving typical problems on the perpendicularity of a straight line and a plane.

If a line is perpendicular to two intersecting lines lying in a plane, then it is perpendicular to this plane.

Let us be given a plane α. There are two straight lines in this plane p And q, intersecting at a point ABOUT(Fig. 1). Straight A perpendicular to a straight line p and straight q. According to the sign, straight A perpendicular to the plane α, that is, perpendicular to any straight line lying in this plane.

3. Mathematics tutor website()

1. Formulate a sign of perpendicularity between a line and a plane.

2. Given a circle with center at a point ABOUT. Straight MO perpendicular to the plane of the circle. Prove that the line MO perpendicular to any radius of the circle.

3. In a triangle ABC height drawn CH. Straight MA perpendicular to the plane ABC. Is the line perpendicular? CH plane AMV?

4. Direct MA perpendicular to the plane of the square ABCD. Find the length of the segments MS,M.B., M.D., if the side of the square is equal a, AM = b.

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Attention! Preview The slides are for informational purposes only and may not represent all the features of the presentation. If you are interested this work, please download the full version.

Class: 10.

Basic tutorial: Geometry 10-11: basic and profile levels/ L.S. Atanasyan et al. - M.: Education, 2009.

The lesson is accompanied by a presentation, a test done in Microsoft Excel for computer testing of students' knowledge ( Annex 1), training module of the Federal Center for Information and Educational Resources ( Appendix 2), consisting of 5 tasks of varying difficulty levels. All tasks of this module are parameterized, which allows you to create individual tasks. The tasks are designed to develop problem solving skills using the sign of perpendicularity of a line and a plane. To work with the training module, you must install special program, she is in Appendix 3. The presentation for the lesson includes independent work on the topic being studied. Thus, the amount of material offered is excessive, which allows it to be dosed and varied depending on the level of preparedness of the class.

Lesson type: lesson in creative application of knowledge.

Form: workshop on solving key problems.

Time spending: 45 minutes.

Place of the lesson in the section: Lesson 4.

Goals:

Educational:

• “discover” the concepts of perpendicular and inclined to a plane;
• develop skills:
  see configurations that satisfy specified conditions;
  apply the definition of a line perpendicular to a plane, the sign of perpendicularity of a line and a plane to proof problems;
• develop skills in solving basic problems on the perpendicularity of a straight line and a plane.

Educational:

• develop spatial imagination, logical thinking;
• develop students’ independence and creative attitude to completing tasks;
• organize an understanding of the results obtained from studying the topic and ways to achieve them.

Educational:

• bring up:
  will and perseverance to achieve final results when solving problems;
  information culture and communication culture.

Methods: partially search, research.

Forms of organization of activities: frontal, group, individual, independent work.

Equipment: computer class, multimedia projector, screen, computer presentation on the topic, test (Appendix 1), cards for individual work (Slide 9), cards with theory questions, electronic educational resources with a practical parameterized task (Appendix 2).

During the classes

Organizational moment - checking the readiness of the class for the lesson.

I. Motivational and orientation part.

1. Updating knowledge.

– Today we continue to work on the topic “Perpendicularity of a line and a plane.” In previous lessons, we “discovered” the definition of a line perpendicular to a plane, the sign of perpendicularity of a line and a plane, and analyzed the simplest problems. As homework, each of you received a sheet with theory questions, you were asked to prepare answers to these questions.

Let's check how you coped with this task.

A frontal survey is underway. (slides 6-8).

The results of the oral work are summed up and the students’ answers are evaluated.

2. Statement of the educational task.

Today we will continue to develop the ability to apply well-known statements in proof problems and in solving standard problems.

1. The next stage of work - two students are called to the board for individual work on cards, with the rest of the students frontal work is carried out using ready-made drawings. Cards for individual work:

Tasks for oral work based on ready-made drawings:

– Guys, in problems 4-6 we are talking about inclined to a plane. What do you think is meant?

Is there an analogy here with the concepts of perpendicular and oblique to a straight line, studied in planimetry?

Students are asked to study slide 10 of the presentation and solve these problems.

2. Work in pairs - solve problems using ready-made drawings.

Solutions are being discussed. Individual student responses are assessed.

The next stage of the lesson is execution practical task on a computer, working with ESM.

III. Reflective-evaluative part.

1. The result of the work During the lesson there is a test in the form of a test.

The lesson is summed up and grades are given.

2. Homework: No. 130, 131, 145, 148. (Instruction: use the sign of perpendicularity of a straight line and a plane).

Geometry. Tasks and exercises on ready-made drawings. 10-11 grades. Rabinovich E.M.

M.: 2014. - 80 p.

The manual is compiled in the form of tables and contains more than 350 tasks. The tasks of each table correspond to a specific topic of the school geometry course for grades 10-11 and are located inside the table in order of increasing complexity.

A high school mathematics teacher knows well how difficult it is to teach students to make visual and correct drawings for stereometric problems.

Due to a lack of spatial imagination, a stereometric task, for which you need to make a drawing yourself, often becomes overwhelming for the student.

That is why the use of ready-made drawings for stereometric problems significantly increases the volume of material covered in the lesson and increases its effectiveness.

The proposed manual is an additional collection of geometry problems for students in grades 10-11 of a general education school and is focused on the textbook by A.V.

Pogorelov "Geometry 7-11". It is a continuation of a similar manual for students in grades 7-9. Format: pdf

(2014, 80 p.) Size:

1.2 MBWatch, download: ; drive.google

Pogorelov "Geometry 7-11". It is a continuation of a similar manual for students in grades 7-9. Rghost djvu

(2014, 80 p.)(2006, 80 p.)

1.3 MB Watch, download:

Download:
Table of contents
Preface 3
Repetition of planimetry course 5
Table 1. Solving triangles 5
Table 2. Area of ​​triangle 6
Table 3. Area of ​​quadrilateral 7
Table 4. Area of ​​quadrilateral 8
Stereometry. 10th grade 9
Table 10.1. Axioms of stereometry and their simplest consequences... 9
Table 10.2. Axioms of stereometry and their simplest consequences. 10
Table 10.3. Parallelism of lines in space. Crossing lines 11
Table 10.4. Parallelism of lines and planes 12
Table 10.5. Sign of parallel planes 13
Table 10.6. Properties of parallel planes 14
Table 10.7. Image of spatial figures on a plane 15
Table 10.8. Image of spatial figures on a plane 16
Table 10.9. Perpendicularity of a line and a plane 17
Table 10.11. Perpendicular and oblique 19
Table 10.12. Perpendicular and oblique 20
Table 10.13. Theorem of three perpendiculars 21
Table 10.14. Theorem of three perpendiculars 22
Table 10.15. Theorem of three perpendiculars 23
Table 10.16. Perpendicularity of planes 24
Table 10.17. Perpendicularity of planes 25
Table 10.18. Distance between crossing lines 26
Table 10.19. Cartesian coordinates in space 27
Table 10.20. Angle between crossing lines 28
Table 10.21. Angle between straight line and plane 29
Table 10.22. Angle between planes 30
Table 10.23. Area of ​​orthogonal projection of a polygon 31
Table 10.24. Vectors in space 32
Stereometry. 11th grade 33
Table 11.1. Dihedral angle. Triangular angle 33
Table 11.2. Straight prism 34
Table 11.3. Correct prism 35
Table 11.4. Correct prism 36
Table 11.5. Inclined prism 37
Table 11.6. Parallelepiped 38
Table 11.7. Constructing prism sections 39
Table 11.8. Regular pyramid 40
Table 11.9. Pyramid 41
Table 11.10. Pyramid 42
Table 11.11. Pyramid. Truncated pyramid 43
Table 11.12. Constructing pyramid sections 44
Table 11.13. Cylinder 45
Table 11.14. Cone 46
Table 11.15. Cone. Truncated cone 47
Table 11.16. Ball 48
Table 11.17. Inscribed and circumscribed ball 49
Table 11.18. Volume of parallelepiped 50
Table 11.19. Prism volume 51
Table 11.20. Pyramid volume 52
Table 11.21. Pyramid volume 53
Table 11.22. Volume of the pyramid. Volume of a truncated pyramid 54
Table 11.23. Volume and lateral surface area of ​​the cylinder..55
Table 11.24. Volume and lateral surface area of ​​the cone 56
Table 11.25. Cone volume. Volume of a truncated cone. The area of ​​the lateral surface of the cone. Lateral surface area of ​​a truncated cone 57
Table 11.26. Volume of the ball. Surface area of ​​the ball 58
Answers, directions, solutions 59

In this article we will talk about the perpendicularity of a line and a plane. First, the definition of a line perpendicular to a plane is given, a graphic illustration and example are given, and the designation of a line perpendicular to a plane is shown. After this, the sign of perpendicularity of a straight line and a plane is formulated. Next, conditions are obtained that make it possible to prove the perpendicularity of a straight line and a plane, when the straight line and the plane are specified by certain equations in a rectangular coordinate system in three-dimensional space. In conclusion, detailed solutions to typical examples and problems are shown.

Page navigation.

Perpendicular straight line and plane - basic information.

We recommend that you first repeat the definition of perpendicular lines, since the definition of a line perpendicular to a plane is given through the perpendicularity of the lines.

Definition.

They say that line is perpendicular to the plane, if it is perpendicular to any line lying in this plane.

We can also say that a plane is perpendicular to a line, or a line and a plane are perpendicular.

To indicate perpendicularity, use an icon like “”. That is, if straight line c is perpendicular to the plane, then we can briefly write .

An example of a line perpendicular to a plane is the line along which two adjacent walls of a room intersect. This line is perpendicular to the plane and to the plane of the ceiling. A rope in a gym can also be considered as a straight line segment perpendicular to the plane of the floor.

In conclusion of this paragraph of the article, we note that if a straight line is perpendicular to a plane, then the angle between the straight line and the plane is considered equal to ninety degrees.

Perpendicularity of a straight line and a plane - a sign and conditions of perpendicularity.

In practice, the question often arises: “Are the given straight line and plane perpendicular?” To answer this there is sufficient condition for perpendicularity of a line and a plane, that is, such a condition, the fulfillment of which guarantees the perpendicularity of the straight line and the plane. This sufficient condition is called the sign of perpendicularity of a line and a plane. Let us formulate it in the form of a theorem.

Theorem.

For a given line and plane to be perpendicular, it is sufficient that the line be perpendicular to two intersecting lines lying in this plane.

You can look at the proof of the sign of perpendicularity of a line and a plane in a geometry textbook for grades 10-11.

When solving problems of establishing the perpendicularity of a line and a plane, the following theorem is also often used.

Theorem.

If one of two parallel lines is perpendicular to a plane, then the second line is also perpendicular to the plane.

At school, many problems are considered, for the solution of which the sign of perpendicularity of a line and a plane is used, as well as the last theorem. We will not dwell on them here. In this section of the article we will focus on the application of the following necessary and sufficient condition for the perpendicularity of a line and a plane.

This condition can be rewritten in the following form.

Let is the direction vector of line a, and is the normal vector of the plane. For straight line a and plane to be perpendicular, it is necessary and sufficient that And : , where t is some real number.

The proof of this necessary and sufficient condition for the perpendicularity of a line and a plane is based on the definitions of the direction vector of a line and the normal vector of a plane.

Obviously, this condition is convenient to use to prove the perpendicularity of a line and a plane, when the coordinates of the directing vector of the line and the coordinates of the normal vector of the plane in a fixed three-dimensional space can be easily found. This is true for cases when the coordinates of the points through which the plane and the line pass are given, as well as for cases when the line is determined by some equations of a line in space, and the plane is given by an equation of a plane of some type.

Let's look at solutions to several examples.

Example.

Prove the perpendicularity of the line and planes.

Solution.

We know that the numbers in the denominators of the canonical equations of a line in space are the corresponding coordinates of the direction vector of this line. Thus, - direct vector .

The coefficients of the variables x, y and z in the general equation of a plane are the coordinates of the normal vector of this plane, that is, is the normal vector of the plane.

Let us check the fulfillment of the necessary and sufficient condition for the perpendicularity of a line and a plane.

Because , then the vectors and are related by the relation , that is, they are collinear. Therefore, straight perpendicular to the plane.

Example.

Are the lines perpendicular? and plane.

Solution.

Let us find the direction vector of a given straight line and the normal vector of the plane in order to check whether the necessary and sufficient condition for the perpendicularity of the line and the plane is met.

The directing vector is straight is