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Presentation "perpendicular straight lines in space." presentation. "Perpendicularity of straight lines and planes in space" Presentation on the topic

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Perpendicularity of straight lines and planes

Perpendicular straight lines in space Two straight lines are called perpendicular because the distance between them is 90 o a b c a  b c  b α

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

A straight line is called perpendicular to a plane because it is perpendicular to any straight line that lies at that plane α a a  α

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

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

Sign of perpendicularity of a straight line and a plane If a straight line is perpendicular to two straight lines that overlap and lie near a plane, then it is perpendicular to that plane. α q Bring: 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) ocremic fall A

α q a p m O Proof: a) zagalny vipadok a 1

Theorem 4 Through any point in space there is a straight line perpendicular to the given area, and only one. α a β M b s Add: 1) ∃ s, s  α , M  s; 2) z - ! Proof: Given: ; M  α

Zavdannya Know: MD A B D M Resolution: Given:  ABC ; MB  BC; MB  BA; MB = BD = a Bring: M B  BD C a a

Zavdannya 128 Bring: O M  (ABC) Given: ABCD - parallelogram; AC ∩ BD = O; M  (ABC); MA = MS, MB = MD A B D C O M Proof:

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

Perpendicular to the wall M A B N α MN  α A  α B  α MA and MV – to the body N  α AN and VN – projections to the body MN – perpendicular to M  α

Theorem about three perpendiculars A straight line is drawn in a plane through the base, perpendicular to its projection onto this plane, perpendicular to the base. A N M α β a Given: a ? ?

The theorem, a reversal of the theorem about three perpendiculars. A straight line is drawn in a plane through the base perpendicular to it, perpendicular to its projection. A N M α β a Given: a ? ?

Cut between a straight line and a plane А Н α β а О φ (а; α) =  AON = φ

Behind the topic: methodical developments, presentations and notes

The presentation on the topic “Perpendicularity of a straight line and a plane” illustrates the theoretical material that is taught in this branch of stereometry.

Presented to a lesson in 10th grade, from geometry to teaching materials: Geometry for 10-11th grade, author L.S. Atanasyan, V.F. Butuzov, S.B. Kadomtsev ta in. This is a lesson in learning new material from wikis...

Slide 2

Perpendicular lines in space Two straight lines in space are called perpendicular (mutually perpendicular), because there is 90° between them. The perpendicularity of straight lines ai b is indicated as follows: ab. On baby 1, the perpendicular lines a and b intersect, and the perpendicular lines a and c intersect. a b c 90° Small 1

Slide 3

If one of two parallel lines is perpendicular to the third line, then the other line is perpendicular to that line. Let us bring the lemma about the perpendicularity of two parallel lines to a third line. Lemma: Proof: Let a || b and ab. Let us see that b  c. Through quite a lot of t.m. space, which does not lie on these straight lines, we will draw straight lines MA and MC, parallel to straight lines a and c. Fragments a  c, then AMC = 90°. Behind the washroom b | a, a z pobudovi a || MA, volume b | | MA. Also, straight lines b and parallel to straight lines MA and MC, where between them there are 90 °. This means that the distance between straight lines b and also equals 90°, i.e. b  c. Small 2 b a C A M c

Slide 4

Parallel lines, perpendicular to a plane A straight line is called perpendicular to a plane because it is perpendicular to any line that lies at that plane. The perpendicularity of the straight line and the plane α is indicated as follows: a  α. If the line is perpendicular to the plane α, then it spans this plane. In fact, if the straight line did not cross the plane α, then it either lay in this plane or was parallel to it. Otherwise, in the plane they would be straight, not perpendicular to the straight line, but, for example, straight lines, parallel to it, which would indicate the appropriate perpendicularity of the straight line and the plane. Well, it’s straight but it crosses the plane?

Slide 5

In picture 3 a straight line a is depicted, perpendicular to the plane α. The environment that inspires us provides a lot of examples that illustrate the perpendicularity of straight lines and planes. The telegraph pole should stand straight without tilting, so that it is perpendicular to the ground. Thus, the self-growing columns were in relation to the surface of the foundation, the lines of the walls were in relation to the area of the foundation, etc. Fig. 3

Slide 6

We present two theorems that establish connections between the parallelism of lines and their perpendicularity to a plane. If one of two parallel lines is perpendicular to a plane, then the other line is perpendicular to that plane. Let's take a look at two parallel lines and a b and a plane α, such that aα. Let's see that b  α. Let’s carry out a straight line x at the plane α (Figure 4). Oskolki a α, then a x. According to the law about the perpendicularity of two parallel lines to the third b  x. In this way, straight line b is perpendicular to any straight line that lies near the plane, then. b  α. Proof: Small. 4 α a b x

Slide 7

If two lines are perpendicular to a plane, they are parallel. Let's look at straight lines a and b, perpendicular to the plane α (Figure 5,a). Let us know that || b. Through Yakus t.M straight b we draw a straight line q parallel to the straight line. Behind the previous theorem q  α. Let us prove that straight line q runs away from straight line b. Tim himself will be informed that a|| b. It is acceptable that straight lines b and q do not coincide. Then in the plane β, which places straight lines b and q through point M, two straight lines pass, perpendicular to straight line c, along which the planes α and β intersect (Figure 5, b). Ale tse is awkward, oh well, and || b. Proof: Small. 5, a α a q Small. 5, b α a M c b b

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Perpendicular

straight to

open spaces

Viznachennya.

Two straight lines are called perpendicular because the smells move under the straight line.

Perpendicular to straight lines on a plane

How many perpendiculars can be drawn to a given line through a given point A, which does not lie on the line, and point B, which lies on the line?

You can pass through the skin point one straight , perpendicular to the data.

Viznachennya. Two straight lines are called perpendicular because the smells move under the straight line.

Show that through any point in space you can draw a straight line perpendicular to the given one.

1. Through directly A and point U let's carry out the area 

2. Through the point U at the square  let's go straight through With, perpendicular to straight line A.

Straight, like

Two straight are called perpendicular as the stench moves under the direct smell.

don't hesitate

lie in the same plane,

are called parallel

Visnovok. Perpendicular to straight lines may lie near different plains.

Know 2 perpendicular straight lines that lie in the same plane and in different planes.

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

Given:

Doc:

Document:

1. Through a sufficient point M, so as not to lie on these lines, we draw MA ||a and MC || With. Because a ┴ c, then AMC = 90˚

3. b|| A.M.

2. b | a (behind the washroom)

a || AM (weekdays)

№ 116 (a) (side 38)

Given:

Doc:

1). DC ┴ B 1 C 1

2). AB ┴ A 1 D 1

Given:

DABC - tetraerd

Doc:

Give the meaning of the perpendicular lines near the space.

2. Formulate your point.

Home improvement:

Theory (page 34, vyvchat)
№ 116 (b), 117

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

DBPOU KK STTT

Mathematics Vikladach

IVANKOVA NADIA PETRIVNA

During the lesson we vibrate...

Know...

Nutrition 1. What straight lines are called perpendicular?

Straight lines in space are called perpendicular, since there is a difference between them 90 0

Powered up 2.

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

Nutrition 3 .

Is a straight line called perpendicular to a plane?

Nutrition 4. Formulate the sign of perpendicularity of a straight line and a plane.

Given: a p, a q

Bring: a α

Nutrition 5 .

What is called rise

from point to plane?

The distance from the point to the plane is called the extension of the perpendicular from the point to the plane

Nutrition 6 .

What is the line between the straight line and

parallel to it?

Nutrition 7 .

What is called the rise between

parallel planes?

Before

Food 8 .

What kind of people are called such that they will meet?



Tip: Intersections are called straight because they do not lie in the same plane

Nutrition 9. How to die stand between straight lines, what to meet?

Vidstan The ancient route appears from any point of one of these straight lines to the plane, which passes through the other straight line, parallel to the first one.

Vidstan between two crisscrossing straight lines There is a high level between two parallel planes, so that they are straight.

Stand straight between the two so that you can meet. ancient dovzhini of the original perpendicular (such a section is united).

Complete the theorem about three perpendiculars

AN – perpendicular to the plane

AB – abduction

VN – projection of AB onto the plane

If a VN, then a AB

Complete the theorem, the reversal theorem about three perpendiculars

Don't lie near the square

And D is perpendicular to the plane?

AB – abduction

D – projection AB onto the plane α

If a AB, then a D

Given: MS ┴ ABC

Know: AS

ABCD – rhombus.

Bring: MO ┴ ABC

Given: DA ┴ ABC

Given: ABCD - parallelogram, MV ┴ ABC

Bring: ABCD - upright

Meals 10:

What do we call a place between a straight line and a plain?

Give the meaning of the dihedral cut.

How does a dihedral cut appear?

Powered 11 : What are the names of the flats?

perpendicular?

Powered 12 : Formulate and convey to the sign

perpendicularity of two planes.

Nutrition 13: What kind of parallelepiped

call him straightforward?

Nutrition 14: Reinvent the authorities of the straightforward

paralelepiped.

Nutrition 15:

Formulate that

prove the theorem about the diagonal

straight-cut

paralelepiped.

Unleash the command:

Given by: ABC D - Straightforward,

MV ⊥ (ABC).

Bring: (AMV) ⊥ (MVS)

At the pyramid DABC in front of the dove ribs: AB = AC = DB=DC =10, ND= D.A. =12. find the position between the straight lines D.A. i Sun.

Trikutniki BDC і ABC equal thighs

D M – height ∆ BDC , D M - median,

AM – median ∆ AB C → AM - Height.

∆ A B.C. = ∆ BDC on three sides D M = AM → ∆ AMD equiosceles