Lesson topic: "Perpendicularity of lines and planes in space"

GBPOU KK STTT

Math teacher

IVANKOVA NADEZHDA PETROVNA

In class, we will be making...

Find...

Question 1. Which lines in space are called perpendicular?

Lines in space are called perpendicular if the angle between them is 90 0

a

b

A

α

Question 2.

Formulate a lemma on the perpendicularity of two parallel lines to a third

a

b

With

M

A

C

α

Question 3 .

Which line is called perpendicular to the plane?

Question 4. Formulate a sign of perpendicularity of a straight line and a plane.

a

Given: a r, a q

Prove: a α

A

l

P

q

Q

p

m

α

L

B

Question 5 .

What is distance

from point to plane?

The distance from a point to a plane is the length of the perpendicular from a given point to a plane

A

a

b

AT

α

Question 6 .

What is the distance between a line and

a plane parallel to it?

a

b

With

α

Question 7 .

What is the distance between

parallel planes?

A

To

Question 8 .

Which lines are called intersecting?

b

α

a



Answer: Crossing lines are lines that do not lie in the same plane.

Question 9. How to measure the distance between intersecting lines?

Distance is equal to the distance from any point of one of these lines to the plane passing through the second line, parallel to the first.

Distance between two intersecting lines is equal to the distance between two parallel planes containing these lines.

Distance between two intersecting lines is equal to the length of their common perpendicular (there is only one such segment).

Prove the three perpendiculars theorem

AN - perpendicular to the plane

AB - oblique

VH - projection of AB onto a plane

If a BH, then a AB

a

Prove a theorem converse to the three perpendiculars theorem

α

A does not lie in a plane

And D is perpendicular to the plane α

AB - oblique

B D is the projection of AB onto the plane α

If a AB, then a B D

a

α

Given: MS ┴ ABC

Find: AC

ABCD is a rhombus.

Prove: MO ┴ ABC

Given: DA ┴ ABC

Given: ABCD - parallelogram, MB ┴ ABC

Prove: ABCD is a rectangle

a

Question 10:

What is the angle between a line and a plane called?

Define a dihedral angle.

How is dihedral angle measured?

a

Question 11 : What planes are called

perpendicular?

Question 12 : Formulate and prove the sign

perpendicularity of two planes.

α

Question 13: What parallelepiped

called rectangular?

Question 14: List the properties of a rectangle

parallelepiped.

Question 15:

Formulate and

prove the diagonal theorem

rectangular

parallelepiped.

Solve the problem:

Given: ABC D - rectangle,

MV ⊥ (ABC).

Prove: (AMV) ⊥ (MVS)

in the pyramid DABC the lengths of the ribs are known: AB=AC= DB=DC =10, BC= DA =12. find the distance between the lines DA and VS.

triangles bdc and ABC isosceles

D M – height ∆ bdc , D M - median,

AM – median ∆ AB C → AM - height.

∆ BUT BC = ∆ bdc on three sides D M = AM → ∆ AMD isosceles

MK – median and height.

MS ⊥ AMD → MS ⊥ MK,

AD ⊥ MK , MK is the common perpendicular of intersecting lines

AD and sun

∆ AVM rectangular, AB=10,

VM=6 , AM=8.

∆ AKM rectangular, AM=8,

AK=6 , MK=2 √ 7.

Solve the problem (according to the figure):

a

Let's draw BE ⊥ AC, CE = EA, since ΔABC is isosceles and the height is also a median.

then by the 3-perpendicular theorem DE ⊥ AC.

Is the statement true?

Straight a is perpendicular to the plane α, and the straight line b

not perpendicular to this plane. Can they

straight a and b be parallel?

b ?

a



Is the statement true?

The line a is parallel to the plane α, and the line b

perpendicular to this plane. Does it exist

a line perpendicular to lines a and b?

b

a

α

Is the statement true?

All lines perpendicular to a given plane

and intersecting the given line lie in the same

planes.

a

b

With

d

α

Is the statement true?

Is it possible to draw three

planes, each two of which are mutually

perpendicular?

SOURCES:

Textbook Geometry Grade 10 AtanasyanL.S. etc. M.: Enlightenment. 2001

http://5terka.com/node/7155

http://vremyazabav.ru/zanimatelno/rebusi/rebusi-slova/82-rebusi-po-matematike.html

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Slides captions:

Perpendicularity of lines and planes

Perpendicular lines in space Two lines are called perpendicular if the angle between them is 90 o a b c a  b c  b α

Lemma If one of two parallel lines is perpendicular to the third line, then the other line is also perpendicular to this line. A C a α M b c Given: a || b, a  c Prove: b  c Proof:

A line is called perpendicular to a plane if it is perpendicular to any line lying in this plane α a a  α

Theorem 1 If one of two parallel lines is perpendicular to a plane, then the other line is also perpendicular to this plane. α x Given: a || a 1 ; a  α Prove: a 1  α Proof: a a 1

Theorem 2 α Prove: a || b Proof: a If two lines are perpendicular to a plane, then they are parallel. β b 1 Given: a  α ; b  α b M c

A sign of perpendicularity of a line and a plane If a line is perpendicular to two intersecting lines lying in a plane, then it is perpendicular to this plane. α q Prove: a  α Proof: a p m O Given: a  p ; a  q p  α ; q  α p ∩ q = O

α q l m O a p B P Q Proof: L a) special case A

α q a p m O Proof: a) general case a 1

Theorem 4 Through any point in space there passes a straight line perpendicular to the given plane, and moreover, only one. α a β М b с Prove: 1) ∃ с, с  α , М  с; 2) with - ! Proof: Given: α ; M  α

Task Find: MD A B D M Solution: Given:  ABC ; MBBC; MBBA; MB = BD = a Prove: М B  BD C a a

Problem 128 Prove: O M  (ABC) Given: ABCD is a parallelogram; AC ∩ BD = O ; M  (ABC); MA = MC, MB = MD A B D C O M Proof:

Task 12 2 Find: AD; BD; AK; B.K. A B D C O K Solution: Given:  ABC – r/s; O - center  ABC CD  (ABC); OK || CD A B = 16  3 , OK = 12; CD = 16 12 16

Perpendicular and inclined M A B N α MN  α A  α B  α

Theorem on three perpendiculars A straight line drawn in a plane through the base of an inclined line perpendicular to its projection onto this plane is perpendicular to the inclined line itself. A N M α β a Given: a  α , AN  α , AM is oblique, a  NM, M  a Prove: a  AM Proof:

The theorem converse to the theorem on three perpendiculars A straight line drawn in a plane through the base of an inclined perpendicular to it is also perpendicular to its projection. A N M α β a Given: a  α , AN  α , AM is oblique, a  AM, M  a Prove: a  HM Proof:

The angle between the straight line and the plane A H α β a O φ (a; α) =  AON = φ

On the topic: methodological developments, presentations and notes

The presentation on the topic "Perpendicularity of a line and a plane" corresponds to the theoretical material studied in this section of solid geometry....

The development of a lesson in grade 10 is presented, in geometry for teaching materials: Geometry for grades 10-11, authors L.S. Atanasyan, V.F. Butuzov, S.B. Kadomtsev and others. This is a lesson in learning new material using ...

Sections: Maths

Lesson Objectives:

• identify the level of mastery of a complex of knowledge and skills to solve problems on a given topic,
• develop spatial imagination, logical thinking, attention and memory,
• educate activity, the ability to listen.

Lesson equipment:

• textbook L.S. Atanasyan and others "Geometry 10-11";
• workbook;
• Personal Computer;
• multimedia projector;
• interactive board;
• author's presentation prepared using Microsoft Power Point ( Attachment 1 )

Lesson structure:

1. Organizing time.
2. Updating the knowledge of students on the topic.
3. Consolidation of previously acquired knowledge and development of skills and abilities to apply this knowledge in solving problems.
4. Summing up the lesson.
5. Homework.

DURING THE CLASSES

1. Organizational moment of the lesson: greeting, checking readiness for the lesson.

2. Updating knowledge obtained by students in the previous lesson:

- the concept of perpendicular lines in space;
- perpendicularity of a straight line and a plane;
– properties of parallel lines perpendicular to the plane.

In order to update knowledge one student goes to the blackboard and writes down the solution to problem No. 119a), the second student is the proof of the theorem on parallel lines perpendicular to the plane.

While they're getting ready, a class frontal poll:

What is the relative position of the two lines in space?
- In what range is the angle between straight lines in space measured?
What lines in space are called perpendicular?
- Formulate a lemma about two parallel lines perpendicular to the third.
– Establish the correct sequence of actions in the proof of the lemma.

After the execution of the online validation.

Teacher: Define the perpendicularity of a line and a plane.

Teacher: Formulate the inverse theorem.

Checking the correctness of the solution of home problem No. 119a (using the equality of triangles).

3. Development of skills and abilities to apply theoretical knowledge to problem solving

1) Oral exercises.

№1 The line AB is perpendicular to the plane, the points M and K belong to this plane. Prove that line AB is perpendicular to line MK.

2) Writing exercises .

№2 In the square ABCD, t.O is the point of intersection of its diagonals. Direct MO is perpendicular to the plane of the square. Prove that MA = MB = MC = MD.

№3 Side AB of parallelogram ABCD is perpendicular to the plane. Find BD if AC = 10 cm.

4. Checking the assimilation of the acquired knowledge during the test

5. Summing up the lesson

Write down a homework assignment: items 15-16, No. 118 No. 120

The presentation "Perpendicular lines in space" is a visual aid for demonstrating educational material when studying the topic of the same name at school. It is difficult to represent figures in space using a blackboard or other standard teacher tools. A presentation is one of the most preferred forms of demonstrating visual material, where it is required to depict bodies in space. When creating a presentation, animation, color representation of figures can be used. Also, the animated presentation contributes to a deeper understanding of the demonstrated processes and transformations, focuses the attention of students on the subject being studied.

During the presentation, students get an idea about lines that are perpendicular in space, an important lemma is formulated and proved about the perpendicularity of a line to both parallel lines when one of them is perpendicular, the solution of the problem is described using the studied material. With the help of the presentation, it is easier for the teacher to form the students' ability to solve geometric problems, to give an idea of ​​the properties of those in space. The material demonstrated during the presentation is easier to understand and remember.

The presentation begins with a reminder of what angle can be formed between two straight lines located on a plane and intersecting with each other. The figure shows a certain plane on which lines a and b are built. When these lines intersect, an angle α is formed. The angle value can be from 0° to 90°. The vertical angles formed by the intersection of the lines are equal, and the adjacent angle is determined by the formula 180°-α. This is theoretical knowledge that the student needs to remember before studying the properties of straight lines perpendicular to space. On the next slide, in order to better demonstrate the mutual position of lines in space, a rectangular parallelepiped ABCDA 1 B 1 C 1 D 1 is shown, on which the edges AA 1 and AB are perpendicular. The definition of perpendicular lines is formulated, which are so called if the angle between them is 90 °. It is also noted that in a rectangular parallelepiped, the lines D 1 C 1 and DD 1 will also be perpendicular to each other. We also recall the notation of perpendicularity of straight lines D 1 C 1 ┴ DD 1 . Next, pairs of lines in the parallelepiped are marked, which will be parallel and perpendicular to each other. It is noted that AA 1 ┴ AD, DD 1 ┴ AD will be perpendicular, and AA 1 and DD 1 are parallel.

The following lemma is presented, which states that if one of the parallel lines is perpendicular to some third line, then the second parallel line will also be perpendicular to it. The wording of the lemma is highlighted for memorization in a frame and with the help of color. The proof of the lemma is demonstrated. The figure shows two parallel lines a and b, as well as a line c, which is known to be perpendicular to a. it is necessary to prove that b and c are also perpendicular. To prove this assertion, an additional point M is constructed, which does not belong to either a or b. A line MA is drawn through this point, parallel to a. MS is also carried out, parallel with. The perpendicularity of a to c means that ∠AMS=90°. From the parallelism of a and b, as well as the parallelism of a to MA, the parallelism of bto MA follows. Since b is parallel to MA, and c is parallel to MC, and the angle ∠AMC=90°, then b is perpendicular to c. The assertion has been proven.

The last slide presents a description of the solution to the problem in which it is required to prove the perpendicularity of the edge of the tetrahedron AM and the line PQ. In the problem, a tetrahedron MABC is given, in which AM is perpendicular to BC. A point P is marked on the edge AB. It is known that AP/AB=2/3. And on the edge Ac, a point Q is marked, which divides the edge in the ratio AQ/QC=2/1. From the relation AQ/QC=2/1 follows the relation Δ/AC=2/3. From the found AQ/AC, the known relation АР/АВ and the fact that the angle ∠А is common, it follows that the triangles ΔARQ and ΔABS are similar. At the same time, from the equality of the angles ∠ARQ=∠ABS, ∠AQP=∠ABC, the lines PQ and BC are parallel. Knowing that the sides Am and BC are perpendicular, and PQ is parallel to BC, using the well-known lemma, we can assert that AM is perpendicular to PQ. Problem solved.

The presentation "Perpendicular Lines in Space" will help the teacher in conducting a geometry lesson at school. Also, visual material is useful for a teacher who conducts training remotely. The presentation can be recommended to a student who independently studies the subject or requires additional material for a deeper understanding.