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

Topic: Perpendicularity of lines and planes

Lesson: Repetition of the theory and solving typical problems on

perpendicularity of a line and a plane (continued)

In this lesson, we will repeat the theory we have covered and continue solution of 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 that 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 the line p and direct q. According to the sign, straight A is perpendicular to the plane α, that is, perpendicular to any line lying in this plane.

3. Math tutor website()

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

2. Given a circle centered 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 held height 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,MB, MD if the side of the square is a, AM = b.

Signs of perpendicularity:

The line is perpendicular to the plane , If _________________________________________

Straight lines are perpendicular , If __________________________________________________

Planes are perpendicular , If ________________________________________________

_______________________________________________________________________________.

Task 1. Construct a ball centered at point A, tangent to a given plane.

Algorithm:

Task 2. Construct a point at a distance of 20 mm from the plane.

Algorithm:

Task 3. Determine the distance from a point to a line.

Algorithm:

Task 4: Complete the missing projection of the triangle if the angle IN straight.

Algorithm:

Task 5 : Construct a square with side BC on a straight line l.

Algorithm:

Task 6 : Complete the projection of the triangle if it is perpendicular to the given plane.

Algorithm:

Questions for self-control of knowledge

In what case is a right angle projected onto the projection plane without distortion?

What is the line of greatest slope?

What is the line of greatest slope in a plane?

How to determine the angle of inclination of the plane to the horizontal, frontal, profile plane of projections?

How is the sign of perpendicularity of a straight line and a plane formulated from the point of view of elementary geometry?

If a line is known to be perpendicular to a plane, how many lines can be drawn that lie in the plane perpendicular to it?

Which two intersecting lines in the plane must be chosen from the set of lines so that the right angle located between them and the given line is projected onto the projection planes without distortion?

Proceeding from this, formulate a sign of perpendicularity of a straight line and a plane from the point of view of descriptive geometry.

How to construct a perpendicular to the plane in general position on CC?

How to construct a straight line perpendicular to the projecting plane on CC?

How is a right angle projected onto the projection plane between intersecting lines if none of them is parallel to this projection plane?

Formulate a sign of perpendicularity of straight lines in general position.

Formulate a sign of perpendicularity of planes.

Topic 11: Method for replacing projection planes

Four main tasks of descriptive geometry:

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At CC remains unchanged __________________________________________________

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

Page navigation.

Perpendicular 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 lines.

Definition.

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

You can also say that the plane is perpendicular to the line, or the line and the plane are perpendicular.

To indicate perpendicularity, use the icon of the form "". That is, if the line c is perpendicular to the plane , then we can briefly write .

As an example of a straight line perpendicular to a plane, one can cite a straight line along which two adjacent walls of a room intersect. This line is perpendicular to the plane and to the plane of the ceiling. The rope in the gym can also be viewed as a straight line perpendicular to the plane of the floor.

In conclusion of this paragraph of the article, we note that if the line is perpendicular to the plane, then the angle between the line and the plane is considered to be 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 line and plane perpendicular?” To answer it, 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 line and the plane. This sufficient condition is called the sign of perpendicularity of a line and a plane. We formulate it in the form of a theorem.

Theorem.

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

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

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

Theorem.

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

At school, many problems are considered, for the solution of which the sign of perpendicularity of a straight line and a plane, as well as the last theorem, is used. Here we will not dwell on them. 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 directing vector of the straight line a , and is the normal vector of the plane . For the perpendicularity of the line a and the plane 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 directing vector of the line and the normal vector of the 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 in three-dimensional space are easily found. This is true for cases where the coordinates of the points through which the plane and the straight line pass are given, as well as for cases where the straight line is determined by some equations of the straight line in space, and the plane is given by an equation of a plane of some kind.

Let's take a look at a few examples.

Example.

Prove that a line is perpendicular and planes.

Solution.

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

The coefficients at the variables x, y, and z in the general equation of the plane are the coordinates of the normal vector of that plane, i.e., 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, a straight line perpendicular to the plane.

Example.

Are the lines perpendicular? and plane.

Solution.

Let us find the direction vector of the given line and the normal vector of the plane in order to check the fulfillment of the necessary and sufficient condition for the line and the plane to be perpendicular.

Direction vector straight is

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

Due to the lack of spatial imagination, the 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 tasks significantly increases the amount of material considered in the lesson, increases its effectiveness.

The proposed manual is an additional collection of problems in geometry 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, 80s.)

Size: 1.2 MB

Watch, download:drive.google ; Rghost

Format: djvu(2006, 80s.)

Size: 1.3 MB

Download: drive.google

Table of contents
Preface 3
Repetition of the course of planimetry 5
Table 1. Solution of triangles 5
Table 2. Area of ​​triangle 6
Table 3. Area of ​​quadrilateral 7
Table 4. Area of ​​quadrilateral 8
Stereometry. 10 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. Crossed 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 line and plane 17
Table 10.10. Perpendicularity of line and plane 18
Table 10.11. Perpendicular and oblique 19
Table 10.12. Perpendicular and oblique 20
Table 10.13. Three perpendiculars theorem 21
Table 10.14. Three perpendiculars theorem 22
Table 10.15. Three perpendiculars theorem 23
Table 10.16. Plane Perpendicularity 24
Table 10.17. Plane Perpendicularity 25
Table 10.18. Distance between crossing lines 26
Table 10.19. Cartesian coordinates in space 27
Table 10.20. Angle between skew lines 28
Table 10.21. Angle between line and plane 29
Table 10.22. Angle between planes 30
Table 10.23. Area of ​​the orthogonal projection of the polygon 31
Table 10.24. Vectors in space 32
Stereometry. 11 class 33
Table 11.1. Dihedral angle. Trihedral 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. Construction of sections of a prism 39
Table 11.8. Correct pyramid 40
Table 11.9. Pyramid 41
Table 11.10. Pyramid 42
Table 11.11. Pyramid. Truncated Pyramid 43
Table 11.12. Sectioning a Pyramid 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 sphere 49
Table 11.18. The volume of the parallelepiped is 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. The volume of the truncated pyramid 54
Table 11.23. The volume and area of ​​the lateral surface of the cylinder..55
Table 11.24. Volume and area of ​​the lateral surface of the cone 56
Table 11.25. Cone volume. The volume of a truncated cone. The area of ​​the lateral surface of the cone. The area of ​​the lateral surface of the truncated cone 57
Table 11.26. The volume of the ball. Ball surface area 58
Answers, instructions, solutions 59

Tasks and exercises on ready-made drawings, grades 10-11, Geometry, Rabinovich E. M., 2006.

Table of contents
Preface.
Repetition of the course of planimetry.
Table 1. Solution of triangles.
Table 2. Area of ​​a triangle.
Table 3. Area of ​​a quadrilateral.
Table 4. Area of ​​a quadrilateral. Stereometry. Grade 10.
Table 10.1. Axioms of stereometry and their simplest consequences.
Table 10.2. Axioms of stereometry and their simplest consequences.
Table 10.3. Parallelism of lines in space. Crossing straight lines.
Table 10.4. Parallelism of lines and planes.
Table 10.5. A sign of parallel planes.
Table 10.6. Properties of parallel planes.
Table 10.7. Image of spatial figures on a plane
Table 10.8. Image of spatial figures on a plane
Table 10.9. Perpendicularity of a line and a plane.
Table 10.10. Perpendicularity of a line and a plane.
Table 10.11. Perpendicular and oblique.
Table 10.12. Perpendicular and oblique.
Table 10.13. Theorem on three perpendiculars.
Table 10.14. Theorem on three perpendiculars.
Table 10.15. Theorem on three perpendiculars.
Table 10.16. Plane perpendicularity.
Table 10.17. Plane perpendicularity.
Table 10.18. Distance between intersecting lines.
Table 10.19. Cartesian coordinates in space.
Table 10.20. Angle between intersecting lines.
Table 10.21. The angle between a line and a plane.
Table 10.22. Angle between planes.
Table 10.23. Area of ​​an orthogonal projection of a polygon
Table 10.24. Vectors in space. Stereometry. Grade 11.
Table 11.1. Dihedral angle. triangular angle.
Table 11.2. direct prism.
Table 11.3. correct prism.
Table 11.4. correct prism.
Table 11.5. tilted prism.
Table 11.6. Parallelepiped.
Table 11.7. Construction of sections of a prism.
Table 11.8. Correct pyramid.
Table 11.9. Pyramid.
Table 11.10. Pyramid.
Table 11.11. Pyramid. Truncated pyramid.
Table 11.12. Construction of a section of a pyramid.
Table 11.13. Cylinder.
Table 11.14. Cone.
Table 11.15. Kohuc. Truncated kohuc.
Table 11.16. Ball.
Table 11.17. Inscribed and circumscribed sphere.
Table 11.18. The volume of the parallelepiped.
Table 11.19. The volume of the prism.
Table 11.20. volume of the pyramid.
Table 11.21. volume of the pyramid.
Table 11.22. volume of the pyramid. The volume of a truncated pyramid.
Table 11.23. The volume and area of ​​the lateral surface of the cylinder.
Table 11.24. The volume and area of ​​the lateral surface of the cone.
Table 11.25. Cone volume. The volume of a truncated cone. The area of ​​the lateral surface of the cone. The area of ​​the lateral surface of a truncated cone.
Table 11.26. The volume of the ball. The surface area of ​​a sphere. Answers, directions, solutions

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