Repetition of the theory and solution of typical problems on the perpendicularity of a line and a plane (continuation). Repetition of the theory and solution of typical problems on the perpendicularity of a line and a plane (continuation) Topic: Perpendicularity of lines and planes

A straight line intersecting a plane is called perpendicular to this plane if it is perpendicular to any straight line in the plane passing through the point of intersection of this line and the plane.

Figure 131 shows a straight line a, perpendicular to the plane a.

T.2.9. If a straight line intersecting a plane is perpendicular to two straight lines in this plane passing through the intersection point, then it is perpendicular to the plane.

This theorem is called the criterion of the perpendicularity of a line and a plane or the theorem of two perpendiculars.

Figure 132 shows straight line a, perpendicular to straight line c and passing through the point of intersection of plane a and straight line a and lying in plane a. It can be argued that.

The next two theorems talk about the relationship between parallelism and perpendicularity of lines and planes.

T.2.10. If a plane is perpendicular to one of two parallel straight lines, then it is perpendicular to the other.

T. 2.11. Two straight lines perpendicular to the same plane are parallel.

Figure 133 shows such lines a and and the plane a, which are mentioned in Theorems 2.10 and 2.11.

The angle between a straight line and a plane is the angle between this straight line and its projection onto the plane.

Figure 134 shows the plane a and the straight line a, which

crosses it. The straight line a is the projection of the straight line a onto the plane a. Then the angle is the angle between the straight line a and the plane a. The angle between the parallel straight line and the plane is considered equal to zero, and the angle between the perpendicular straight line and the plane is equal to 90 °. Since the straight line a, its projection a on the plane a and the perpendicular to the plane a at the point of its intersection with the straight line a lie in the same plane, the angle between the straight line and the plane complements the angle between this straight line and the perpendicular to the plane to 90 °.

Example. A 10 cm segment intersects the plane, and its ends are at a distance of 3 and 2 cm from the plane. Find the angle between the given line segment and the plane.

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

Table of contents
Foreword.
Repetition of the planimetry course.
Table 1. Solving triangles.
Table 2. Area of a triangle.
Table 3. Area of the quadrangle.
Table 4. Area of the quadrangle. Stereometry. Grade 10.
Table 10.1. Stereometry axioms and their simplest consequences.
Table 10.2. Stereometry axioms and their simplest consequences.
Table 10.3. Parallelism of straight lines in space. Crossed straight lines.
Table 10.4. Parallelism of lines and planes.
Table 10.5. Parallelism of planes.
Table 10.6. Parallel plane properties.
Table 10.7. Drawing spatial figures on a plane
Table 10.8. Drawing spatial figures on a plane
Table 10.9. Perpendicularity of a straight line and a plane.
Table 10.10. Perpendicularity of a straight line and a plane.
Table 10.11. Perpendicular and oblique.
Table 10.12. Perpendicular and oblique.
Table 10.13. Three perpendicular theorem.
Table 10.14. Three perpendicular theorem.
Table 10.15. Three perpendicular theorem.
Table 10.16. Perpendicularity of the planes.
Table 10.17. Perpendicularity of the planes.
Table 10.18. Distance between crossed lines.
Table 10.19. Cartesian coordinates in space.
Table 10.20. The angle between crossing lines.
Table 10.21. The angle between a straight line and a plane.
Table 10.22. The angle between the planes.
Table 10.23. Polygon orthogonal projection area
Table 10.24. Vectors in space. Stereometry. Grade 11.
Table 11.1. Dihedral angle. Triangular corner.
Table 11.2. Straight prism.
Table 11.3. Correct prism.
Table 11.4. Correct prism.
Table 11.5. Inclined prism.
Table 11.6. Parallelepiped.
Table 11.7. Construction of sections of the prism.
Table 11.8. Correct pyramid.
Table 11.9. Pyramid.
Table 11.10. Pyramid.
Table 11.11. Pyramid. Truncated pyramid.
Table 11.12. Building a section of the pyramid.
Table 11.13. Cylinder.
Table 11.14. Cone.
Table 11.15. Kohuc. Truncated kohуc.
Table 11.16. Ball.
Table 11.17. Inscribed and described ball.
Table 11.18. The volume of the parallelepiped.
Table 11.19. The volume of the prism.
Table 11.20. The volume of the pyramid.
Table 11.21. The volume of the pyramid.
Table 11.22. The volume of the pyramid. Truncated pyramid volume.
Table 11.23. 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. The volume of the cone. Truncated cone volume. The area of the lateral surface of the cone. The area of the lateral surface of the truncated cone.
Table 11.26. The volume of the ball. Ball surface area. Answers, directions, solutions

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6.1 Determination of the perpendicularity of a line and a plane

The idea of straight lines or, rather, of segments perpendicular to the plane, is given by vertically standing pillars (they are perpendicular to the surface of the earth), a stretched cord on which the lamp hangs (it is perpendicular to the ceiling), table legs (they are perpendicular to the floor). The vertical jamb of the door is perpendicular to the floor, and the lower edge of the door, adjacent to the floor, is perpendicular to the jamb in all positions of the door (Fig. 73, a). This property determines the perpendicularity of a straight line and a plane.

Definition. A straight line is called perpendicular to the plane if it intersects this plane and is perpendicular to any straight line in this plane passing through the intersection point (Fig. 73, b).

Rice. 73

They also say that the plane is perpendicular to a straight line or that they are mutually perpendicular. For mutually perpendicular straight line a and plane a, the designations a ⊥ α or α ⊥ a are used.

A segment or ray is perpendicular to a plane if it lies on a straight line perpendicular to this plane. If a segment is perpendicular to the plane and its end lies in this plane, then it is called perpendicular to this plane.

6.2 Perpendicular and oblique

A segment that has one common point with a plane - the end of the segment, but not perpendicular to this plane, is called inclined to the plane.

Let from one point A, not lying in the plane a, a perpendicular AB and an oblique AC are drawn (Fig. 74). The segment BC is called the projection of the inclined AC on the plane α.

Rice. 74

Perpendicular AB is shorter than inclined AC, i.e. AB< АС. Действительно, в прямоугольном треугольнике ABC катет АВ короче гипотенузы АС. Итак, перпендикуляр короче наклонной, если они проведены из одной и той же точки к одной плоскости.

It can be said this way: the perpendicular AB from point A to the plane α is the shortest of the segments connecting point A with points of the plane α.

The property of the perpendicular to be the shortest line is a characteristic property. This means that the converse is also true: if AB is the shortest segment from point A to the plane α, then AB is the perpendicular to the plane α.

Proof. Let us prove this by contradiction. Suppose AB is not perpendicular to α. Then a straight line a passes through point B in the plane α, which is not perpendicular to AB (Fig. 75). Let us drop the perpendicular AM from A to the line a. In a right-angled triangle AVM, the AM leg is less than the hypotenuse AB: AM< АВ. Но тогда отрезок АВ не будет кратчайшим из всех отрезков, идущих из точки А до плоскости а. Получили противоречие. Следовательно, АВ ⊥ α.

Rice. 75

The length of the perpendicular, lowered from the highest point of the object to its base, measures the height of the object. So, the height of the pyramid is the length of the perpendicular lowered from the top of the pyramid to the plane of its base, as well as the perpendicular itself (in Figure 76, a, b - this is a segment RO).

Rice. 76

6.3 About the meaning of the perpendicular

The perpendicular to the plane plays a very important role, and besides the fact that it is the shortest among all segments from a given point to points on the plane. Let us also explain its meaning. The position of the plane in space can be set by specifying a line perpendicular to it and the point at which it intersects this line.

The most important property of the perpendicular is that the plane is located symmetrically relative to it. What does it mean? All rays lying in a given plane form equal angles with it - right angles, but this is not the case for an inclined plane (Fig. 77, a). When rotating around the perpendicular, the plane aligns with itself: the wheel must be mounted on the axis so that its plane is perpendicular to the axis. A rectangle with a side perpendicular to the plane can be rotated around that side, and the other side will slide along the plane. This can be clearly seen on a properly hinged door. If its edge is not vertical, the door will not open freely and will rub against the floor.

Rice. 77

Taking examples from physics, it can be noted that the pressure of a liquid or gas on the wall of the vessel is directed perpendicular to the wall, just as the pressure of the load on the support is directed perpendicular to it (Fig. 77, b and 78, a).

Rice. 78

The perpendicular to the surface appears in the laws of reflection and refraction of light. So, the law of reflection says: "The incident ray and the reflected ray are located in the same plane with the perpendicular to the surface of the mirror at the point of incidence and form equal angles with it." "Angle of incidence" and "angle of reflection" are the angles between the specified perpendicular and the incident beam and the reflected beam (Fig. 78, b).

But the main meaning of the perpendicular is its role in technology and in our entire life.

We can be said to be surrounded by perpendiculars: the legs of the table are perpendicular to the floor, the edge of the cabinet is perpendicular to the wall, etc.

The vertical is perpendicular to the horizontal plane. The verticality is checked with a plumb line (see photo). Perpendicularity plays a major role in construction: interfloor ceilings are laid perpendicular to the pillars of the building frame.

As we will see later, the parallelism of the planes is associated with the presence of common perpendiculars. The perpendicularity and parallelism of straight lines and planes is an essential element in construction, so that the doctrine of perpendiculars and parallels can be called the foundations of "construction geometry".

Questions for self-control

What is the difference between perpendicular to plane and inclined to plane?
What definition of a perpendicular to a plane do you know?
What is the meaning of the perpendicular to the plane?

Signs of perpendicularity:

Straight line perpendicular to plane , if _________________________________________

Straight lines are perpendicular , if __________________________________________________

The planes are perpendicular , if ________________________________________________

_______________________________________________________________________________.

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

Algorithm:

Objective 2. Draw a point at a distance of 20 mm from the plane.

Algorithm:

Objective 3. Determine the distance from a point to a straight line.

Algorithm:

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

Algorithm:

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

Algorithm:

Problem 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 called the line of greatest slope?

How is the line of greatest slope located in the plane?

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

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

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

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

Based on 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 the CP?

How to build a straight line perpendicular to the projection plane on the CP?

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

Formulate a criterion for the perpendicularity of straight lines in general position.

Formulate the sign of the perpendicularity of the planes.

Topic 11: Method of replacing projection planes

The four main tasks of descriptive geometry are:

_____________________________________________________________________________

Remains unchanged at CC __________________________________________________

________________________________________________________________________________

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, the designation of perpendicular lines and a plane is shown. After that, a criterion for the perpendicularity of a straight line and a plane was formulated. Further, conditions are obtained that make it possible to prove the perpendicularity of a straight line and a plane, when a straight line and a plane are given by some equations in a rectangular coordinate system in three-dimensional space. In the 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 straight lines, since the definition of a straight line perpendicular to a plane is given through the perpendicularity of straight lines.

Definition.

They say that straight line perpendicular to the plane if it is perpendicular to any straight 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 it can be written briefly.

An example of a straight line perpendicular to a plane is 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. A rope in a gym can also be viewed as a straight line segment perpendicular to the plane of the floor.

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

The perpendicularity of a straight line and a plane is a sign and conditions for 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 straight 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 the perpendicularity of a line and a plane. Let us formulate it in the form of a theorem.

Theorem.

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

The proof of the sign of perpendicularity of a straight line and a plane can be found in the 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 the two parallel lines is perpendicular to the plane, then the second line is perpendicular to the plane.

The school considers many problems, for the solution of which the criterion 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 as follows.

Let be is the direction vector of the straight line a, and is the normal vector of the plane. For the perpendicularity of the straight 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 straight line and a plane is based on the definitions of the direction vector of the straight line and the normal vector of the plane.

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

Let's consider solutions of several examples.

Example.

Prove the perpendicularity of a straight line and plane.

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 direction vector of this straight line. Thus, - directing vector of a straight line .

The coefficients for the variables x, y and z in the general equation of the 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 the line and the plane.

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

Example.

Is the line 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 perpendicularity of the line and the plane.

The direction vector of the straight line is an