Perpendicularity of a line and a plane definition: A straight line is called perpendicular to a plane if it is perpendicular to any line lying in this one. Perpendicularity of a line and a plane

GPOU "Usinsk Polytechnic College"

Public lesson in geometry

Topic: “Perpendicularity of a line and a plane.”

Completed by: mathematics teacher Melnikova E.A.

Usinsk, 2016

Lesson type: Lesson-seminar

Goals lesson :

Summarize, consolidate and systematize students’ knowledge on this topic, the ability to apply this knowledge when solving problems; show the practical significance of the material being studied; study the connection between the relations of parallelism and perpendicularity in space; show interdisciplinary connections.

Foster a culture of oral and writing, contribute to the education of aesthetic taste, instill interest in the subject of mathematics.

Develop spatial and logical thinking.

Equipment for the lesson: cards with the names Theorists, Practitioners, Researchers, group assignments, PC, projector.

Lesson plan.

I. Student organization.

Students are offered cards with the names Theorists, Practitioners, Researchers and are divided into 3 groups.

II. Setting goals and objectives for the lesson.

They say that mathematics is an uninteresting science, that mathematics is a dry science, that it can only be discussed in the mathematics classroom, in class. No, life proves the opposite: mathematics is all around us. Listen to what Roman Bukharaev writes about this in his poem “Geometry of Herbs.”

Unfulfilled mathematician, wanderer,
Look around, surprised a hundred times:
In the grasses there is a cut of thistle - a pentagon,
And the cross-section of oregano is a square.
Everything in the world will seem new
Under the char, whose top is covered in snow:
The catchment area is triangular at its base.
In a blooming alpine meadow!
Where is the circle?
Near the needle rose.
Where the heavenly meadow is rocky,
I see birch trees playing with the wind
Triangular-rhombic sheet.

But I agree that mathematics is an exact science, requiring clear definitions and proof of facts. And so now I propose to move from the lyrics to practice.

You have studied a very important topic in geometry: “Perpendicularity of a line and a plane.” As a result of studying this topic you should:

know the definitions of perpendicular lines and a line perpendicular to a plane.

be able to form and prove theorems (direct and inverse) about parallel lines, lines perpendicular to a plane, a sign of perpendicularity of a line and a plane, a theorem about a line perpendicular to a plane.

Solve problems like 119, 121, 126, 128, 131 (study “Geometry 10-11”, author L.S. Atanasyan)

The teacher introduces the objectives of the lesson.

III. Consolidation of knowledge and skills.

During the lesson there will be 3 groups: “Theoreticians”, “Practicians”, “Researchers”.

The teacher gives the groups a task prepared on sheets. Indicates the order of grading.

Before the groups start working, there is a frontal readiness check.

What could it be like mutual arrangement 2 straight lines in space? (Straight lines can intersect, cross and be parallel.)

Which two lines are called parallel? (Parallel lines are called straight lines , which lie in oneplanes and either coincide or do not intersect.)

Which two lines are called intersecting? ( Lines are called intersecting if one of the lines lies in a plane and the other lies in this plane.the plane intersects at a point not belonging to the first line.)

If the angle between two straight lines is 900, what are they called? (Perpendicular lines)

Which line is called perpendicular to the plane? (A line is called perpendicular to a plane if it is perpendicular to any line lying in this plane.

Is the statement true:

a) Any line perpendicular to a plane intersects this plane? (right)
b) Any line intersecting a plane is perpendicular to this plane? (wrong)
c) If a line is not perpendicular to a given plane, then it does not intersect this plane? (wrong)

Line a is parallel to line b and does not intersect the plane? Can a straight line B be perpendicular to a plane? Justify your answer. (it cannot be, because if straight line b is perpendicular to the plane, then straight line a is also perpendicular to the plane, which is impossible, because according to the condition, straight line a does not intersect the plane, therefore it is parallel to the plane)

1. Assignments for the “Theoreticians” group.

Prove the lemma about the perpendicularity of two parallel lines to a third line.

Lemma. If one of two parallel lines is perpendicular to the third line, then the other line is perpendicular to this line.

Given:a ‖ b, a ⊥ c

Prove: b ⊥ c

Proof:

Through a point M in space that does not lie on these lines, we draw lines MA and MC, parallel to lines a and c, respectively. Since a ⊥ c, then ∠ AMC = 90o.

By condition, b ‖ a, and by construction, a ‖ MA, therefore b ‖ MA.

So, straight lines b and c are parallel to straight lines MA and MC, respectively, the angle between them is 90°, i.e. b ‖ MA, c ‖ MC, the angle between MA and MC is 90°

This means that the angle between straight lines b and c is also equal to 90°, that is, b ⊥ c. The lemma is proven.

Prove theorems (direct and converse) about parallel lines, lines perpendicular to a plane.

Theorem:(straight line) If one of two parallel lines is perpendicular to a plane, then the other line is perpendicular to this plane.

Write on the board and in notebooks:

D ano: a ‖ a1, a ⊥ α

Prove that a1 ⊥ α

Proof:

Let us draw some straight line x in the α plane, i.e. x ∊ α. Since a ⊥ α, then a ⊥ x.

By the lemma about the perpendicularity of two parallel lines to the third, a1 ⊥ x.

Thus, the line a1 is perpendicular to any line lying in the plane α, i.e., a1 ⊥ α. The theorem is proven.

Theorem:(reverse) If two lines are perpendicular to a plane, then they are parallel.

Given: a ⊥ α, b ⊥ α

Prove that a ‖ b

Proof:

Through some point M of line b we draw a line b1 parallel to line a.

M ∊ b, M ∊ b1, b1 ‖ a. By the previous theorem, b1 ⊥ α.

Let us prove that line b1 coincides with line b. Thus we will prove that a ‖ b. Let us assume that the lines b1 and b do not coincide. Then in the plane β containing lines b and b1, two lines pass through the point M, perpendicular to the line c along which the planes α and β intersect. But this is impossible, therefore, a ‖ b, i.e. b ∊ β, b1 ∊β, α β=c (impossible)→ a ‖ b.

Generate and analyze the proof of the sign of perpendicularity of a line and a plane.

Sign of 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 the plane itself

At the end of the “Theorists” group, the teacher gives the floor to the student with historical information"Hanging a straight line."

To carry out long straight sections (when laying a highway or railway, power lines, etc.) a method called straight hanging is used, which consists of using all poles about 2 m long, pointed at one end so that they can be stuck into the ground. If it is necessary to draw a straight line between two points A and B, the position of which is given, then first milestones are placed at these points; then an intermediate pole C is installed between them so that poles A and C cover pole B. It is necessary that all poles stand vertically. The correct vertical direction is checked using a plumb line. A plumb line is a cord with a small weight attached to the end. It would seem that everything is clear in this simple procedure for hanging a straight line. But here too there are many questions to think about, and the answers to them are provided by studying our course and other disciplines. Firstly, why do all the plumb lines of the world look at the center of the Earth, and from the point of view of geometry, they define a straight line perpendicular to its surface? Secondly, the pole must be parallel to the plumb line, and then it will also be perpendicular to the surface of the Earth. Thus, all milestones are perpendicular to the surface of the Earth and, therefore, parallel to each other.

This method is called hanging a straight line on the ground. The word "hanging" is a derivative of the word "milestone".

2. Group tasks "Practices".

Show the application of the theory in solving problems No. 126, 127, 128,131 (p. 42 of the lesson “Geometry 10-11 author L.S. Atanasyan)

3. Group tasks "Researchers".

Study the relationship between the relations of parallelism and perpendicularity in space. Check using the table.

Given a line a, perpendicular to the plane α, and a line b. Indicate the relative position of lines a and b:

If b is parallel, then......

If b is perpendicular, then......

If b is parallel or belongs to , then.....

If b is perpendicular, then......

Given a straight line a, perpendicular to the plane α, and a plane.

If parallel, then......

If perpendicular, then......

If a is parallel or a belongs to , then.....

If perpendicular, then......

Give examples of the environment around us that illustrate the perpendicularity of a straight line and a plane.

At the end of the group work, students give examples of the location of lines in physics problems (interdisciplinary communication)

Think about the power of pressure. How is it directed? (Perpend. to the surface plane).

Body on a horizontal surface. How does the force of gravity mg act on any body? What is its direction?

The body is immersed in liquid. It is subject to a buoyant force. What is its direction?

IV. Summing up the lesson. Grading.

V. Homework.

P.15 - 16, questions 1, 2 (p. 57), No. 116, 118.

In planimetry, the construction of a perpendicular is based on the fact that it connects a given point and a point symmetrical with it relative to the line under consideration. If we want to formulate the concept of a perpendicular to a plane, then we can take any point lying outside this plane, reflect this point in a given plane, as in a mirror, and connect this point with its reflection; then we get a perpendicular to the plane. It should be noted, however, that in the case of reflection relative to a straight line, the whole matter came down to bending the plane along a given straight line, i.e., to movement, albeit produced in space. Reflection in a plane is no longer reduced to movement. Therefore, the presentation of the question of a perpendicular to a plane is more complicated than the corresponding presentation of the question of a perpendicular to a line in planimetry; it is based on the following known to the reader

Definition. A line is called perpendicular to a plane if it is perpendicular to any line lying in this plane.

Since the angle between two intersecting straight lines is equal, by definition, to the angle between intersecting straight lines parallel to the data, then straight line a (Fig. 337), perpendicular to all straight lines of plane K passing through the point of intersection of straight line a with plane K, will also be perpendicular to plane K Indeed, it forms a right angle with any line in the plane since it is perpendicular to the line b drawn in this plane through a point parallel to b.

In reality, there is a much simpler test for the perpendicularity of a line and a plane. A line perpendicular to two intersecting lines of a plane is perpendicular to that plane.

Proof. Let in Fig. 338 line a is perpendicular to two intersecting lines lying in the X plane. By virtue of the above remark, we can, without loss of generality, assume that line a passes through the point of intersection of the lines type. It is required to prove that straight line a is perpendicular and to any straight plane, due to the same remark, we can assume that the straight line passes through the point . Let us make the following auxiliary constructions: on straight line a we take an arbitrary point M and a point M on the continuation on the other side of the plane H at a distance from the point Three straight lines in the plane X we intersect any line c that does not pass through the points of intersection we denote respectively P, Q, R Let's connect points M and M with points P, Q, R. The triangles are equal, since they are rectangular, the legs are equal in construction, and the leg is common; this means that their hypotenuses are also equal: (you can even more simply notice that MR - MR, like oblique ones with equal projections). The segments MQ, MQ are also equal. This means that the triangles MPQ and MPQ are equal (on three sides). From here we conclude that the triangles MQR are congruent and they have equal angles between the equal sides MQ and MQ and the common side QR: (the corresponding angles in equal triangles). Now we can see that triangles are equal to three sides). Thus, the angles MMUR are equal, and since they are adjacent, each of them is right. The statement has been proven.

A perpendicular plane can be drawn to any straight line.

In fact, let's take an arbitrary straight line and at any point draw two perpendiculars to it (lying in any two planes drawn through this straight line). A plane passes through them, like through two intersecting lines. According to the previous one, this straight line serves as a perpendicular to this plane.

From the above reasoning, the conclusion also follows: all lines perpendicular to a given line at one of its points lie in the same plane perpendicular to this line.

At any point of the plane, you can also restore a perpendicular to it.

To do this, it is enough to draw two straight lines lying in this plane through a given point in a plane, and then construct at the same point two planes perpendicular to the drawn lines. Having a common point, these two planes will intersect along a straight line, which will be simultaneously perpendicular to the two intersecting lines in the plane and, therefore, perpendicular to the plane itself.

In this lesson we will repeat the theory and prove the theorem that indicates the perpendicularity of a line and a plane.
At the beginning of the lesson, let's remember the definition of a line perpendicular to a plane. Next, we will consider and prove the theorem that indicates the perpendicularity of a line and a plane. To prove this theorem, recall the property of the perpendicular bisector.
Next, we will solve several problems on the perpendicularity of a line and a plane.

Topic: Perpendicularity of a line and a plane

Lesson: Sign of perpendicularity of a line and a plane

In this lesson we will repeat the theory and prove theorem-test of perpendicularity of a line and a plane.

Definition. Straight A is called perpendicular to the plane α if it is perpendicular to any line lying in this plane.

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

Proof.

Let us be given a plane α. There are two intersecting lines in this plane p And q. Straight A perpendicular to a straight line p and straight q. We need to prove that the line A is perpendicular to the plane α, that is, that line a is perpendicular to any line lying in the plane α.

Reminder.

To prove it, we need to recall the properties of the perpendicular bisector to a segment. Perpendicular bisector R to the segment AB- this is the locus of points equidistant from the ends of the segment. That is, if the point WITH lies on the perpendicular bisector p, then AC = BC.

Let the point ABOUT- point of intersection of the line A and plane α (Fig. 2). Without loss of generality, we will assume that the straight lines p And q intersect at a point ABOUT. We need to prove the perpendicularity of the line A to an arbitrary line m from the α plane.

Let's draw through the point ABOUT direct l, parallel to the line m. On a straight line A put aside the segments OA And OB, and OA = OB, that is, the point ABOUT- the middle of the segment AB. Let's make a direct P.L., .

Straight R perpendicular to a straight line A(from the condition), (by construction). Means, R AB. Dot R lies on a straight line R. Means, RA = PB.

Straight q perpendicular to a straight line A(from the condition), (by construction). Means, q- perpendicular bisector to a segment AB. Dot Q lies on a straight line q. Means, QA =QB.

Triangles ARQ And VRQ equal on three sides (RA = PB, QA =QB, PQ-common side). So the angles ARQ And VRQ are equal.

Triangles AP.L. And BPL equal in angle and two adjacent sides (∠ ARL= ∠VRL, RA = PB, P.L.- common side). From the equality of triangles we obtain that AL =B.L..

Consider a triangle ABL. It is isosceles because AL =BL. In an isosceles triangle, the median LО is also the height, that is, a straight line LО perpendicular AB.

We got that straight A perpendicular to a straight line l, and therefore direct m, Q.E.D.

Points A, M, O lie on a line perpendicular to the plane α, and the points O, V, S And D lie in the α plane (Fig. 3). Which of the following angles are right angles: ?

Solution

Let's consider the angle. Straight JSC is perpendicular to the plane α, which means it is a straight line JSC perpendicular to any line lying in the α plane, including the line IN. Means, .

Let's consider the angle. Straight JSC perpendicular to a straight line OS, Means, .

Let's consider the angle. Straight JSC perpendicular to a straight line ABOUTD, Means, . Consider a triangle DAO. A triangle can only have one right angle. So the angle DAM- is not direct.

Let's consider the angle. Straight JSC perpendicular to a straight line ABOUTD, Means, .

Let's consider the angle. This is an angle in a right triangle BMO, it cannot be straight, since the angle MOU- straight.

Answer: .

In a triangle ABC given: , AC= 6 cm, Sun= 8 cm, CM- median (Fig. 4). Through the top WITH a direct line was drawn SK, perpendicular to the plane of the triangle ABC, and SK= 12 cm Find KM.

Solution:

Let's find the length AB according to the Pythagorean theorem: (cm).

By property right triangle middle of the hypotenuse M equidistant from the vertices of the triangle. That is SM = AM = VM, (cm).

Consider a triangle KSM. Straight KS perpendicular to the plane ABC, which means KS perpendicular CM. So it's a triangle KSM- rectangular. Let's find the hypotenuse KM from the Pythagorean theorem: (cm).

1. Geometry. Grades 10-11: textbook for general students educational institutions(basic and profile levels) / I. M. Smirnova, V. A. Smirnov. - 5th edition, corrected and expanded - M.: Mnemosyne, 2008. - 288 pp.: ill.

Tasks 1, 2, 5, 6 p. 57

2. Define the perpendicularity of a line and a plane.

3. Indicate a pair in the cube - an edge and a face that are perpendicular.

4. Point TO lies outside the plane of an isosceles triangle ABC and equidistant from the points IN And WITH. M- middle of the base Sun. Prove that the line Sun perpendicular to the plane AKM.

Maintaining your privacy is important to us. For this reason, we have developed a Privacy Policy that describes how we use and store your information. Please review our privacy practices and let us know if you have any questions.

Collection and use of personal information

Personal information refers to data that can be used to identify or contact a specific person.

You may be asked to provide your personal information at any time when you contact us.

Below are some examples of the types of personal information we may collect and how we may use such information.

What personal information do we collect:

• When you submit an application on the site, we may collect various information, including your name, telephone number, address Email etc.

How we use your personal information:

• Collected by us personal information allows us to contact you and inform you about unique offers, promotions and other events and upcoming events.
• From time to time, we may use your personal information to send important notices and communications.
• We may also use personal information for internal purposes such as auditing, data analysis and various studies in order to improve the services we provide and provide you with recommendations regarding our services.
• If you participate in a prize draw, contest or similar promotion, we may use the information you provide to administer such programs.

Disclosure of information to third parties

We do not disclose the information received from you to third parties.

Exceptions:

• If necessary - in accordance with the law, judicial procedure, legal proceedings, and/or based on public requests or requests from government agencies on the territory of the Russian Federation - disclose your personal information. We may also disclose information about you if we determine that such disclosure is necessary or appropriate for security, law enforcement, or other public importance purposes.
• In the event of a reorganization, merger, or sale, we may transfer the personal information we collect to the applicable successor third party.

Protection of personal information

We take precautions - including administrative, technical and physical - to protect your personal information from loss, theft, and misuse, as well as unauthorized access, disclosure, alteration and destruction.

Respecting your privacy at the company level

To ensure that your personal information is secure, we communicate privacy and security standards to our employees and strictly enforce privacy practices.

Perpendicularity of a line and a plane.