The theorem on the line perpendicular to the plane. Proof. The topic of the lesson is "The theorem on the line perpendicular to the plane

A straight line perpendicular to a plane is perpendicular to any straight line in this plane. Based on the right angle projection theorem, and its essence is as follows:

in rectangular projection, a right angle is projected in full size (right angle) only if one of its sides is parallel to the projection plane, and the other is not perpendicular to this plane,

as straight lines of the general position plane, it is most convenient to use its level lines.

Therefore, drawing a perpendicular to the plane, it is necessary to take two such straight lines in this plane: the horizontal and the frontal.

The projections of a straight line perpendicular to the plane in the complex drawing are perpendicular to the corresponding projections of its level lines, i.e. if a straight line is perpendicular to the plane, then its horizontal projection should be perpendicular to the horizontal projection of the horizontal, and its frontal projection should be the frontal projection of the frontal (Fig. 67) or the corresponding traces of the plane (Fig. 68).

In fig. 69 shows the plane in general position  ( a b) to which you want to draw a perpendicular line.

Rice. Fig. 68

Rice. 69

Draw a horizontal line in this plane h(through points 1,3) and frontal v(through points 1,4) (Fig. 69).

Then from point 1 we draw a straight line n perpendicular to the horizontal and front of the plane as follows:

n "  h " n ""  h ""

Constructed line n(n ",n "") is the required perpendicular to the plane.

1. Perpendicular planes

Two planes are mutually perpendicular if one of them passes through the perpendicular to the other. The construction of such planes can be done in two ways:

1) the plane is drawn through a perpendicular to the other;

2) the plane is drawn perpendicular to a straight line belonging to another plane.

In fig. 70 shows a straight line in general position l and the plane in general position ( a b). It is required to build through a straight line l plane perpendicular to plane.

Rice. 70

To solve the problem, it is necessary through some point of this straight line, for example, a point M, draw a perpendicular to the plane defined by intersecting straight lines a and b.

Draw in the plane  horizontal h and frontal v(fig. 70).

Further from the point M taken on a straight line l, we lower the perpendicular n, using the above statement: n "h ";n ""v "", i.e. the horizontal projection of the perpendicular will be perpendicular to the horizontal projection of the horizontal, and its frontal projection will be perpendicular to the frontal projection of the frontal (Fig. 70).

The plane  ( l n) passing through the straight line n, will be perpendicular to the plane.

1. Perpendicular straight lines

Two straight lines are perpendicular if and only if a plane perpendicular to the other can be drawn through each of them.

In fig. 71 depicts a straight line l general position, to which you want to draw a perpendicular line.

Rice. 71

Through point A straight l we construct a plane perpendicular to it ( h v):

l "  h "; l ""  h ""(fig. 71).

Any line lying in the plane  will also be perpendicular to this line l... Therefore, we draw in this plane an arbitrary straight line t, at which we take an arbitrary point, for example, the point V(fig. 71).

By connecting the dots A and V lying in the plane, we get a straight line n perpendicular to this line l(fig. 71).

REVIEW QUESTIONS

What is called the line of greatest inclination of the plane?

How to determine the angle of inclination of the plane to the frontal plane of the projections?

How is the mutual perpendicularity of a line and a plane displayed in a complex drawing?

Formulate necessary and sufficient conditions for the perpendicularity of two straight lines in general position.

Under what conditions are two planes in general position perpendicular to each other?

How to draw a plane perpendicular to a given straight line?

How to draw a perpendicular from a point to a straight line in general position?

How to build mutually perpendicular planes?

In this lesson, we will consider and prove the theorem about the only line perpendicular to the plane.
At the beginning of the lesson, we will formulate the studied theorem on the existence of a single straight line passing through a given point and perpendicular to this plane. To prove it, we first consider and prove the assertion about the existence of a plane perpendicular to a given line. After proving the theorem, we will consider several corollary problems on the topic under study.

Topic: Perpendicularity of a line and a plane

Lesson: Theorem about a line perpendicular to a plane

In this lesson we will look at and prove the theorem on the only line perpendicular to the plane.

Statement

A plane perpendicular to this straight line passes through any point in space.

Proof(see fig. 1)

Let us be given a straight line a and point M... Let us prove that there is a plane γ that passes through the point M and which is perpendicular to the straight line a.

Through a straight line a draw the planes α and β so that the point M belongs to the plane α. Planes α and β intersect in a straight line a... In the plane α through the point M draw a perpendicular MN(or R) to straight a,. In the plane β from the point N restore the perpendicular q to straight a... Direct R and q intersect, let the plane γ pass through them. We get that the straight line a perpendicular to two intersecting straight lines R and q from the plane γ. Hence, according to the criterion of perpendicularity of a straight line and a plane, the straight line a perpendicular to the plane γ.

Theorem

A straight line perpendicular to a given plane passes through any point in space, and, moreover, only one.

Proof.

Let a plane α and a point M(see fig. 2). It is necessary to prove that through the point M is the only straight line With perpendicular to the plane α .

Let's draw a straight line a in the plane α (see Fig. 3). According to the statement proved above, through the point M one can draw the plane γ perpendicular to the straight line a... Let it be straight b- the line of intersection of the planes α and γ.

In the plane γ through the point M let's draw a straight line With perpendicular to the straight line b.

Straight With perpendicular b by construction, straight With perpendicular a(since straight a is perpendicular to the plane γ, and hence to the straight line With, lying in the plane γ). We get that the straight line With perpendicular to two intersecting straight lines from the plane α. Hence, according to the criterion of perpendicularity of a straight line and a plane, the straight line With perpendicular to the plane α. Let us prove that such a straight line With the only one.

Suppose there is a straight line With 1 passing through the point M and perpendicular to the plane α. We get that straight lines With and from 1 perpendicular to the plane α. So straight With and from 1 are parallel. But by construction, straight With and from 1 intersect at the point M... We got a contradiction. Hence, there is a single straight line passing through the point M and perpendicular to the plane α, as required.

Prove that if two planes α and β are perpendicular to the line a, then they are parallel.

Proof:

Let's draw a straight line With parallel straight a... By the lemma, if one of the two parallel lines intersects the plane, then the other line also intersects the plane. Straight a intersects the planes α and β by condition. Means straight With intersects the plane α at some point A and the plane β at point B.

Straight a perpendicular to the planes α and β, and hence the line parallel to it With perpendicular to the planes α and β.

Suppose that the planes α and β intersect. Dot M is the common point of the planes α and β. But then in a triangle AMV injection MAV is equal to 90 ° and the angle AVM equals 90 °, which is impossible. Hence, the assumption that the planes α and β intersect was incorrect. Hence, the planes α and β are parallel.

Prove that only one plane perpendicular to the given line passes through any point in space.

Proof:

Let a straight line be given a and point M... According to the statement, there is a plane γ passing through the point M perpendicular to the straight line a... Let us prove its uniqueness.

Suppose that there is a plane γ 1 passing through the point M perpendicular to the straight line a... Two planes γ and γ 1 are perpendicular to the same straight line a, and hence the planes γ and γ 1 are parallel (as we proved in Problem 1). But the point M belongs to the plane γ and γ 1. We got a contradiction. This means that only one plane passes through any point in space, perpendicular to a given straight line a, as required to prove.

So, we have proved the theorem on the line perpendicular to the plane. In the next lesson, we will look at solving problems with such straight lines.

1. Geometry. Grades 10-11: a textbook for students of educational institutions (basic and profile levels) / I. M. Smirnova, V. A. Smirnov. - 5th edition, revised and supplemented - M.: Mnemosina, 2008. - 288 p. : ill.

2. Geometry. Grade 10-11: Textbook for general education educational institutions/ Sharygin I.F. - M .: Bustard, 1999 .-- 208 p.: Ill.

3. Geometry. Grade 10: Textbook for educational institutions with in-depth and specialized study of mathematics / E. V. Potoskuev, L. I. Zvalich. - 6th edition, stereotype. - M.: Bustard, 008 .-- 233 p. : ill.

1. Geometry. Grades 10-11: a textbook for students of educational institutions (basic and profile levels) / I. M. Smirnova, V. A. Smirnov. - 5th edition, revised and supplemented - M .: Mnemozina, 2008. - 288 p .: ill.

Tasks 15, 16, 17 p. 58

2. Is it true that the line is perpendicular to those lying in this plane:

a) two sides of a triangle

b) two sides of the trapezoid

c) two diameters of the circle.

3. Prove that two different straight lines perpendicular to it can be drawn through any point of a straight line in space.

4. Direct a,b, With lie in the plane α. Straight m perpendicular to straight lines a and b but not perpendicular With... What is mutual arrangement direct a and b?

Repeat paragraph 1, paragraphs 15-18, all the properties and theorems are written in your notebook, study paragraph 18, write down the theorem on a line perpendicular to a plane in your notebook.

Two straight lines in space are called perpendicular if the angle between them is 90o.

Perpendicular lines can intersect and can be crossed. Lemma. If one of the two parallel lines is perpendicular to the third line, then the other line is perpendicular to this line. Definition. A straight line is called perpendicular to the plane if it is perpendicular to any straight line lying in the plane. They also say that the plane is perpendicular to the straight line a.

rice. 38

If the straight line a is perpendicular to the plane, then it obviously intersects this plane. Indeed, if the straight line a did not intersect the plane, then it would lie in this plane or would be parallel to it. But in either case, there would be straight lines in the plane that are not perpendicular to the straight line but, for example, straight lines parallel to it, which is impossible. Hence, the straight line a intersects the plane.

The relationship between the parallelism of straight lines and their perpendicularity to the plane.

A sign of perpendicularity of a straight line and a plane.

Remarks.

Through any point in space, there is a plane perpendicular to a given straight line, and, moreover, the only one. A straight line perpendicular to a given plane passes through any point in space, and, moreover, only one. If two planes are perpendicular to a straight line, then they are parallel.

Explore the answers to the questions:

In space, perpendicular lines can intersect and can be crossed. (Yes, for example a cube.) If one of the two parallel lines is perpendicular to the third line, then the other line is parallel to this line. (No, it is perpendicular.) A straight line is called perpendicular to a plane if it is perpendicular to any straight line lying in this plane. (No, because, by condition, the lines can lie in this plane.) If one of the two parallel lines is perpendicular to the plane, then the other line is also parallel to the plane. (No, it is perpendicular.) If a straight line is perpendicular to two intersecting straight lines lying in a plane, then it is perpendicular to this plane. (Yes, by sign.) If a straight line is perpendicular to a plane, then it is perpendicular to two sides of the triangle lying in this plane. (Yes.) If a line is perpendicular to the plane, then it is perpendicular to the two sides of the square. (Not.)

In the tetrahedron ABCD (Figure 1) BCD = ACD = 90 °. Is it true that in the figure the edges AB, AC, BC are perpendicular to CD? (Yes.),

Given: ∆ ABC, VM AB, VM VS, D AC.

What is symmetry. Symmetry in geography. Symmetry in Geology. Natural objects... Examples of symmetric distribution. Types of symmetry. Cylinder symmetry. Symmetry of the external shape of the crystal. Symmetry in biology. Discrete symmetry. Symmetry in nature. Symmetry is a fundamental property of nature. Symmetry in physics. Symmetrical shapes. Man, many animals and plants have bilateral symmetry.

"The condition of perpendicularity of a straight line and a plane" - Theorem about a straight line perpendicular to a plane. The angle between a straight line and a plane. Direct MA and MS. Let us prove that the line a is perpendicular to an arbitrary line m. Oblique properties. Theorem on two parallel lines. Theorems establishing a connection between parallelism. Line a is perpendicular to the plane of the ASM. Three perpendicular theorem. Build plan. Theorem on two straight lines perpendicular to the plane.

"Methods for constructing sections" - Formation of skills and abilities for constructing sections. Memo. Consider four cases of constructing sections of a parallelepiped. Cutting plane. Internal design method. Construction of sections of polyhedra. The trace is called the line of intersection of the section plane and the plane of any face of the polyhedron. The box has six faces. Construct sections of the tetrahedron. Trace method. Working with disks.

"Consequences from the axioms of stereometry" - Elements of a cube. Plane. Draw a straight line. Which planes does the point belong to? Geometry slides. Find the line of intersection of the planes. Solution. Various planes. Planimetry axioms. Independent work... Assertions. Build a cube image. Planimetry. The existence of a plane. Planes. Proof. Lines that intersect at a point. Stereometry axioms and some of their consequences.

"Define Dihedral Angles" - Faces of a parallelepiped. Where can you see the theorem of three perpendiculars. Task. Let's draw the beam. Plane M. The point on the edge can be arbitrary. A figure formed by a straight line a and two half-planes. Dihedral angles in pyramids. Perpendicular, oblique and projection. Point K. The angle at the lateral edge of a straight prism. Definition and properties. Rhombus. The ends of the segment. Triangular corner property. Perpendicular planes.

"Parallelepiped" - "Salzburg parallelepiped". Study the properties of geometric shapes using algebra. A tetrahedron can be inscribed in a parallelepiped. Parallelepiped. Rectangular parallelepiped. Diagonal properties rectangular parallelepiped... Development of geometry. The diagonals of a straight parallelepiped are calculated using the formulas. This is how the box looks in a flat pattern. The parallelepiped is symmetrical about the midpoint of its diagonal.

Lecture on the topic "Theorem on the line perpendicular to the plane"

Let's remember them: The first theorem The criterion for the perpendicularity of a line and a plane

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

And two theorems on parallel lines direct theorem. If one of two parallel lines is perpendicular to the plane, then the other line is perpendicular to this plane.

AND converse theorem. If two lines are perpendicular to the plane, then they are parallel. We have already discussed the proof of these theorems.

On-screen text:

A sign of perpendicularity of a straight line and a plane. If a straight line is perpendicular to two intersecting straight lines lying in a plane, then it is perpendicular to this plane.

The text is added to the screen:

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

The text is added to the screen:

Converse theorem. If two lines are perpendicular to the plane, then they are parallel.

Task.

Prove that a plane perpendicular to the given line passes through any point in space.

To solve the problem, consider the line a, and an arbitrary point in space, the point M. Let us prove that there is a plane passing through the point M and perpendicular to the line a.

For the proof, we draw two planes α and β containing the line a, since this is their common line, hence the line and their line of intersection.

In the plane β through the point M we draw a straight lineb perpendicular to the linea ... let these lines intersect at point O.

In the plane α, draw a straight lineWith passing through point O and perpendicular to the straight linea .

By the theorem on the existence of a plane, namely through two intersecting linesv and With it is possible to draw a plane and, moreover, only one.

Consider the planeγ ( gamma ) passing through the straight linesWith and b .

Plane γ( gamma ) will be the desired plane, since line a is perpendicular to two intersecting lines b and c

On the screen the text of the problem: Prove that a plane perpendicular to the given line passes through any point in space.

On-screen drawing

The drawing is updated on the screen and a solution point is added.

Proof:

Let us draw α, β: a, M

The drawing and the proof point are updated on the screen 2)

Proof:

We will carry out b: b , b , ba, b a = O

The drawing is updated on the screen and a proof point is added 3)

Proof:

Let's spend with: With , With, With a

Proof item added 4)

Added proof clause 5)

a ⊥.

This problem demonstrates the existence of a plane perpendicular to a given straight line. Consider a theorem stating the existence and uniqueness of a straight line perpendicular to a given plane.

Consider the plane α and an arbitrary point in space - point A.

Let us prove that through point A there passes the only straight line perpendicular to the given plane.

1,2) So, we draw in the plane α an arbitrary straight linem... Let's build a planeso that it passes through point A perpendicular to the linem.

3,4) Let the plane α and β intersect along the straight linen... In the plane β, through point A we draw a straight line p, perpendicular to the straight linen.

5) Direct T , perpendicular to the planeβ , so it is perpendicular to any straight line in this plane, that is, the straight lineT perpendicular to a straight lineR .

6) Then the straight line pm and n lying in the planeα , therefore, based on the perpendicularity of a straight line and a plane, the straight linep perpendicular to the planeα .

7) It is important to understand that there can be only one such straight line. If two straight lines passed through point A, for example, another straight linep 1 perpendicular to the plane α. But two straight lines perpendicular to one plane are parallel, which contradicts our assumption. Thus, only one straight line perpendicular to this plane passes through a point in space.

This statement in geometry is called the theorem on the line perpendicular to the plane.

On-screen text:

A straight line perpendicular to a given plane passes through any point in space, and, moreover, only one.

On-screen drawing

Proof:

Let's carry out m: m

Consider β: βА

αβ = n

We will carry out p : p , A R , pm .

Clause is added to the proof 6)

The drawing and the proof point are updated on the screen:

e noun

Task

Through vertices A and B of rectangle ABCD 1 and BB 1 1 AB and AA 1 A D D= 25 cm, AB = 12 cm, AD= 16 cm.

Solution 1) Since straight AA 1 perpendicular to two intersecting straight linesADand AB lying in the plane of the rectangle, then the sign of the perpendicularity of the straight line to the plane AA 1 D.

2) Straight BB 1 parallel to straight line AA 1 hence, by the theorem, the line BB 1 perpendicular to the plane ABCD, and is perpendicular to any straight line lying in this plane, that is, BB 1 perpendicular to line BD... So triangle B 1 in D rectangular.

3) From right triangle BAD by the Pythagorean theorem the square of the hypotenuseBDis equal to the sum of the squares of the legs AB andAD and BD equals 20 cm.

4) By the Pythagorean theorem from a right-angled triangle B 1 in D... Squared leg B 1 B is equal to the difference of the squares of the hypotenuse B 1 Dand the famous legBD, and the leg is 15 cm.

On the screen the text of the task. Through vertices A and B of rectangle ABCDparallel straight lines AA are drawn 1 and BB 1 not lying in the plane of the rectangle. It is known that AA 1 AB and AA 1 A D... Find BB 1 if B 1 D= 25 cm, AB = 12 cm, AD= 16 cm

On-screen text and drawing:

Solution:

Item 2 is added to the solution) drawing is updated

Item 3 is added to the decision)

1. : by the Pythagorean theorem

V D=

Item 4) is added to the decision and then the answer

: by the Pythagorean theorem

Answer: 15 cm.

Consider a proof problem.

Line a is perpendicular to the plane α and perpendicular to the lineb b||

Let's call the intersection point of the straight line and the plane-point M.

1,2) Note on the linea some pointNnot lying on a straight lineb... Through a point that does not lie on this line, you can draw the only line parallel to the given one. Let this straight line beb 1 .

3) Through point Nlet's draw a straight line from 1 .

4) Through point M in the plane α we draw a straight line with parallel to the straight line with 1 .

5) Through two intersecting lines with 1 and b 1 the plane β can be drawn according to the theorem on the existence of a plane.

6) The straight line a is perpendicular by the condition of the plane α, which means it is perpendicular to the straight line c lying in the plane, but c is parallel to the straight line c 1 , hence the straight line a is perpendicular to the straight line c 1 .

7.8) Similarly, line a is perpendicular to linebby condition, straightbparallel to the straight lineb 1 , therefore, the straight line a is perpendicular to the straight lineb 1 ... So straight line a, on the basis of the perpendicularity of the line and the plane, is perpendicular to the plane β.

9) The planes α and β are perpendicular to the straight line and, therefore, they are parallel.

10) Direct bparallel to the straight lineb 1 , so it is parallel to the plane β, and parallel to the plane α.

On the screen, the text of the task:

Problem 3. Line a is perpendicular to the plane α and perpendicular to the lineb not lying in this plane. Prove thatb ||

Given: a, a b

Prove that || b

Proof:

Note N: .

b 1 : b 1

The blueprint is updated on the screen and a proof point is added:

Let's carry out with 1: with 1

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Let's spend with: with

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Added proof clause 6):

a ⊥α

Added proof clause 7) 8)

ab .

Proof 9 added)

Added proof clause 10)