Table 10.9 perpendicularity of a line and a plane solution. Perpendicular line and plane, sign and conditions of perpendicularity of a line and plane

6.1 Determination of 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 poles (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 a 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 the line, or that they are mutually perpendicular. For mutually perpendicular lines a and plane a, the designations a ⊥ α or α ⊥ a are used.

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

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

Rice. 74

The perpendicular AB is shorter than the oblique 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.

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

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

Proof. Let us prove this by contradiction. Assume that AB is not perpendicular to α. Then a straight line a passes through the point B in the plane α, not perpendicular to AB (Fig. 75). Let us drop the perpendicular AM from A to the line a. In a right triangle AVM, the leg AM 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 is the 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 the segments from a given point to the points of the plane. Let's explain its meaning. The position of a plane in space can be specified by specifying a straight line perpendicular to it and the point at which it intersects this straight line.

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

Rice. 77

Taking examples from physics, it can be noted that the pressure of a liquid or gas on the wall of a vessel is directed perpendicular to the wall, just as the pressure of a load on a 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. Thus, the law of reflection states: "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 indicated perpendicular and the incident ray and the reflected ray (Fig. 78, b).

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

We are, so to speak, surrounded by perpendiculars: the legs of a table are perpendicular to the floor, the edge of a cabinet is perpendicular to the wall, etc.

The vertical is perpendicular to the horizontal plane. Verticality is checked with a plumb line (see photo). Perpendicularity plays a major role in construction: interfloor floors 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 lines and planes is an essential element in construction, so the doctrine of perpendiculars and parallels can be called the foundations of "building geometry".

Questions for self-control

1. What is the difference between perpendicular to a plane and oblique to a plane?
2. What is the definition of a perpendicular to a plane?
3. What is the meaning of perpendicular to a plane?

Back forward

Attention! The slide preview is for informational purposes only and may not represent the full extent of the presentation. If you are interested in this work, please download the full version.

Class: 10.

Basic Tutorial: Geometry 10-11: basic and profile levels / L.S. Atanasyan and others - M .: Education, 2009.

The lesson is accompanied by a presentation, a test made in Microsoft Excel for computerized testing of students' knowledge ( Attachment 1), a training module of the Federal Center for Information and Educational Resources ( Appendix 2), consisting of 5 tasks of different difficulty levels. All tasks of this module are parameterized, which allows you to create individual tasks. Tasks are designed to develop the skills of solving problems using the sign of perpendicularity of a straight line and a plane. To work with the training module, you need to install a special program, it is located in Annex 3. In the presentation for the lesson there is an independent work on the topic under study. Thus, the amount of the proposed material is redundant, which allows it to be dosed, vary depending on the level of preparedness of the class.

Lesson type: a lesson in the creative application of knowledge.

Conduct form: workshop for solving key problems.

Time spending: 45 minutes.

Place of the lesson in the section: 4 lesson.

Goals:

Tutorials:

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

Developing:

• develop spatial imagination, logical thinking;
• to develop the independence of students and a creative attitude to the implementation of tasks;
• organize the comprehension of the results of the study of the topic and ways to achieve them.

Educational:

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

Methods: partially exploratory, research.

Forms of organization of activity: 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, EER with a practical parameterized task (Appendix 2).

During the classes

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

I. Motivational and orienting part.

1. Actualization of knowledge.

– Today we continue to work on the topic “Perpendicularity of a line and a plane”. In the past lessons, we “discovered” the definition of a straight line perpendicular to a plane, a sign of perpendicularity of a straight line and a plane, and analyzed the simplest tasks. 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.

There is a face-to-face survey. (slides 6-8).

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

2. Statement of the learning task.

Today we will continue to form the ability to apply known statements in problems of proof and in solving typical problems.

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

Tasks for oral work on finished drawings:

- Guys, in tasks 4-6 we are talking about inclined to the 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 invited to study slide 10 of the presentation and solve these problems.

2. Work in pairs - tasks are solved according to ready-made drawings.

Solutions are being discussed. Individual student responses are evaluated.

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

III. Reflective-evaluative part.

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

The results of the lesson are summed up, marks are given.

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

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 but perpendicular to the line p and direct q. According to the sign, straight but 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.

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 be 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 direction 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. In this way, - 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 an

Signs of perpendicularity:

The line is perpendicular to the plane , if _________________________________________

Straight lines are perpendicular , if __________________________________________________

Planes are perpendicular , if ________________________________________________

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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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