Perpendicularity in space can have:

1. Two straight lines

3. Two planes

Let's consider these three cases in turn: all the definitions and statements of theorems related to them. And then we will discuss a very important theorem about three perpendiculars.

Perpendicularity of two lines.

Definition:

You can say: they opened America to me too! But remember that in space everything is not quite the same as on a plane.

On a plane, only such lines (intersecting) can be perpendicular:

But perpendicularity in space of two lines can be even if they do not intersect. Look:

A line is perpendicular to a line, although it does not intersect it. How so? We recall the definition of the angle between lines: to find the angle between skew lines and, you need to draw a line through an arbitrary point on the line a. And then the angle between and (by definition!) will be equal to the angle between and.

Remembered? Well, in our case, if the lines and turn out to be perpendicular, then the lines and should be considered perpendicular.

To be completely clear, let's look at example. Let there be a cube. And you are asked to find the angle between the lines and. These lines do not intersect - they intersect. To find the angle between and, draw.

Due to the fact that - a parallelogram (and even a rectangle!), It turns out that. And due to the fact that - a square, it turns out that. Well, that means.

Perpendicularity of a line and a plane.

Definition:

Here is the picture:

a line is perpendicular to a plane if it is perpendicular to all-all lines in this plane: and, and, and, and even! And a billion other lines!

Yes, but how then can you generally check perpendicularity in a straight line and a plane? So life is not enough! But fortunately for us, mathematicians saved us from the nightmare of infinity by inventing sign of perpendicularity of a line and a plane.

We formulate:

Check out how great:

if there are only two lines (s) in the plane to which the line is perpendicular, then this line will immediately turn out to be perpendicular to the plane, that is, to all lines in this plane (including some line standing on the side). This is a very important theorem, so we will also draw its meaning in the form of a diagram.

And let's look again example.

Let us be given a regular tetrahedron.

Task: to prove that. You will say: these are two straight lines! What does the perpendicularity of a straight line and a plane have to do with it ?!

But look:

let's mark the middle of the edge and draw and. These are the medians in and. Triangles are regular and.

Here it is, a miracle: it turns out that, as well as. And further, to all straight lines in the plane, and hence, and. Proved. And the most important point was precisely the use of the sign of perpendicularity of a straight line and a plane.

When the planes are perpendicular

Definition:

That is (for more details, see the topic “dihedral angle”), two planes (s) are perpendicular if it turns out that the angle between the two perpendiculars (s) to the line of intersection of these planes is equal. And there is a theorem that connects the concept of perpendicular planes with the concept of perpendicularity in the space of a line and a plane.

This theorem is called

Criterion of perpendicularity of planes.

Let's formulate:

As always, the decoding of the words "then and only then" looks like this:

• If, then passes through the perpendicular to.

• If passes through the perpendicular to, then.

(naturally, here and are planes).

This theorem is one of the most important in stereometry, but, unfortunately, one of the most difficult to apply.

So you need to be very careful!

So the wording is:

And again, deciphering the words "then and only then." The theorem states two things at once (look at the picture):

Let's try to apply this theorem to solve the problem.

A task: a regular hexagonal pyramid is given. Find the angle between the lines and.

Solution:

Due to the fact that in a regular pyramid the vertex falls into the center of the base during projection, it turns out that the line is the projection of the line.

But we know that in a regular hexagon. We apply the three perpendiculars theorem:

And write the answer:

PERPENDICULARITY OF LINES IN SPACE. BRIEFLY ABOUT THE MAIN

Perpendicularity of two lines.

Two lines in space are perpendicular if the angle is between them.

Perpendicularity of a line and a plane.

A line is perpendicular to a plane if it is perpendicular to all lines in that plane.

Plane perpendicularity.

Planes are perpendicular if the dihedral angle between them is equal.

Criterion of perpendicularity of planes.

Two planes are perpendicular if and only if one of them passes through the perpendicular to the other plane.

Three perpendiculars theorem:

Well, the topic is over. If you are reading these lines, then you are very cool.

Because only 5% of people are able to master something on their own. And if you have read to the end, then you are in the 5%!

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13.11.2016 14:35

Test tasks in geometry for the section "Lines and planes in space" 1. Axioms of stereometry. 2. Parallelism of straight lines and planes. 3. Perpendicularity of straight lines and planes. Answers at the end of development

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"Test tasks in geometry for the section "Lines and planes in space" 1st year of SPO"

Section number 3. Lines and planes in space

The subject of stereometry. Basic concepts and axioms of stereometry. Spatial figures.

Parallelism of lines in space. Parallelism of two planes.

Vectors in space. Parallel transfer.

Section of polyhedra.

Perpendicularity of lines, lines and planes.

Perpendicular and oblique.

The angle between a line and a plane. Dihedral angle. Plane perpendicularity.

Axioms of stereometry

Option 1

	1) ABC 2) DBC 3) DAB 4) DAC
	How flat does the point K belong to the pit? 1) ABC and ABD
	Select faithful statements: 1) Any three points lie in the same plane. 2) If the center of the circle and its point lie in a plane, then the whole circle lies in this plane. 3) Only one plane passes through three points on a straight line. 4) A plane passes through two intersecting lines, and moreover, only one. Answer: ______
	Select infidels statements: 1) If three lines have a common point, then they lie in the same plane. 3) Two planes can have only two common points. 4) Three pairwise intersecting lines at different points lie in the same plane. Answer: ______
	Name the line along which the planes A 1 BC and A 1 AD intersect. 1) DC 2) A 1 D 1 3) D 1 D 4) D 1 C
	Name the line along which the planes DCC 1 and A 1 AD intersect. 1) DC 2) A 1 D 1 3) D 1 D 4) D 1 C
	Lines AB and CD intersect. A plane is drawn through the line AB. Name the line of intersection of this plane with the plane BCD. 1) AC 2) AB 3) BC 4) BD
	Lines AB and CD intersect. A plane is drawn through points B and D. Name the line of intersection of this plane with the plane ACD. 1) AC 2) AB 3) BC 4) BD

Option 2

	The point P lies on the line MN. Name the plane to which point R belongs. 1) ABC 2) DBC 3) DAB 4) DAC
	What planes does point F belong to? 1) ABC and ACD
	Select faithful statements: 1) Any four points lie in the same plane. 2) Only one plane passes through a line and a point not lying on it. 3) If three points of a circle lie in a plane, then the whole circle lies in this plane. 4) Two planes can have only one common point. Answer: ______
	Select infidels statements: 1) Two circles having a common center lie in the same plane. 3) Three vertices of a triangle belong to the same plane. 4) A plane passes through two parallel lines, and moreover, only one. Answer: ______
	Name the line along which the planes DCC 1 and A 1 BC intersect. 1) DC 2) A 1 D 1 3) D 1 D 4) D 1 C
	Name the line along which the planes ABC and C 1 CB intersect. 1) BC 2) B 1 C 1 3) A 1 B 4) B 1 B
	Lines AB and CD intersect. A plane is drawn through the line CD. Name the line of intersection of this plane with the plane ABC. 1) CD 2) AD 3) BC 4) BD
	Lines AB and CD intersect. A plane is drawn through points A and D. Name the line of intersection of this plane with the BCD plane. 1) AC 2) AD 3) BC 4) BD

Option 1

	Points M, P, K are the midpoints of the edges DA, DB, DC of the tetrahedron DABC. Name the line parallel to the plane FBC. 1) MR 2) RK 3) MK 4) MK and RK
	АВСDA 1 B 1 C 1 D 1 is a rectangular parallelepiped. Which of the lines is parallel to the plane A 1 B 1 C 1 ? 1) a 2) b 3) p 4) m
	In the tetrahedron DABC, VC = KS, DP = PC. Which plane of the face is parallel to the straight line RK? 1) DAB 2) DBC 3) DAC 4) ABC
	Select faithful statements: 1) Two lines in space are called parallel if they do not intersect. 2) If one of two parallel lines is parallel to a plane, then the other line is either parallel to it or lies in this plane. 3) There is a line that lies in a plane and is parallel to a line intersecting the given plane. 4) Intersecting lines do not have common points. Answer: ______
	1) a || n 2) a || b 3) b || c 4) a || c
	faithful statements: 1) Straight CD and MN crossing. 2) The lines AB and MN lie in the same plane. 3) Lines CD and MN intersect. 4) Straight AB and CD crossing. Answer: ______
	1) a and b – intersecting lines 2) a and b – parallel lines 3) a and b – intersecting lines
	Determine the relative position of the lines. 1) a and b – intersecting lines 2) a and b – parallel lines 3) a and b – intersecting lines
	Triangles ABK and ABF are arranged so that lines AB and FK intersect. How are lines AK and BF located?
	In the tetrahedron DABC AB = BC = AC = 20; DA \u003d DB \u003d DC \u003d 40. Through the middle of the rib AC, a plane parallel to AD and BC. Find the perimeter of the section. Answer: ____

Parallelism of lines and planes

Option 2

	Points M, P, K are the midpoints of the edges DA, DB, DC of the tetrahedron DABC. Name the line parallel to plane FAB. 1) MR 2) RK 3) MK 4) MK and RK
	АВСDA 1 B 1 C 1 D 1 is a rectangular parallelepiped. Which of the lines is parallel to the plane A 1 AD? 1) a 2) b 3) p 4) m
	In the tetrahedron DABC AM = MD, AN = NB. Which face plane is line MN parallel to? 1) DAB 2) DBC 3) DAC 4) ABC
	Select faithful statements: 1) Parallel lines do not have common points. 2) If a line is parallel to a given plane, then it is parallel to any line lying in this plane. 3) If a line is parallel to the line of intersection of two planes and does not belong to any of them, then it is parallel to each of these planes. 4) There is a parallelepiped in which all corners of the faces are acute. Answer: ______
	Points A, B, C and D are the midpoints of the edges of a rectangular parallelepiped. Name the parallel lines. 1) a || n 2) a || b 3) b || c 4) a || c
	Points A and D are the midpoints of the edges of the parallelepiped. Select faithful statements: 1) Lines CD and MN intersect. 2) Straight AB and MN crossing 3) Straight lines AB and CD are parallel. 4) Lines AB and MN intersect Answer: ______
	Determine the relative position of the lines. 1) a and b – intersecting lines 2) a and b – parallel lines 3) a and b – intersecting lines
	Points A and B are the midpoints of the edges of the parallelepiped. Determine the relative position of the lines. 1) a and b – intersecting lines 2) a and b – parallel lines 3) a and b – intersecting lines
	Two isosceles triangles ABC and ABD with a common base AB are located so that the point C does not lie in the plane ABD. Determine the relative position of the lines containing the medians of the triangles drawn to the sides BC and BD. 1) they are parallel 2) they intersect 3) they intersect
	In the tetrahedron DABC AB = BC = AC = 10; DA \u003d DB \u003d DC \u003d 20. Through the middle of the edge BC, a plane parallel to AC and BD. Find the perimeter of the section. Answer: ____

Option 1

	Through side AB of triangle ABC a plane is drawn perpendicular to side BC. Determine the type of triangle with respect to the angles.
	Triangle ABC is regular, O is the center of the triangle. The distance from point M to vertex A is 3. Find the height of the triangle. Answer: ____
	ABCD - parallelogram; Find the perimeter of the parallelogram. 1) 20 2) 25 3) 40 4) 60
	A plane α parallel to BC is drawn through the vertex A of triangle ABC. The distance from BC to the plane α is 12. Find the distance from the point of intersection of the medians of the triangle ABC to this plane. 1) 8 2) 6 3) 12 4) 18
	The height of the rhombus is 12. The point M is equidistant from all sides of the rhombus and is at a distance equal to 8 from its plane. What is the distance of point M to the sides of the rhombus? Answer: ____
	Select faithful statements: 2) Two lines perpendicular to the same plane are parallel. 3) The length of the perpendicular is less than the length of the oblique drawn from the same point. 4) Two intersecting lines can be perpendicular to the same plane. Answer: ______
	Segment AB rests with ends A and B on the faces of a right dihedral angle. The distances from points A and B to the edge are 1, and the length of the segment AB is 3. Find the length of the projection of this segment onto the edge.
	In the DABC tetrahedron, AO cuts off BC at point E; Find.
	Rectangle ABCD and parallelogram BEMC are arranged so that their planes are mutually perpendicular. Find the angle MCD.

Perpendicularity of lines and planes

Option 2

	Through the side AD of the parallelogram ABCD, a plane is drawn perpendicular to the side DC. Determine the type of triangle ABC. 1) acute 2) rectangular 3) obtuse
	Triangle ABC is regular, O is the center of the triangle. The height of the triangle is 3. Find the distance from point M to the vertices of the triangle. Answer: ____
	ABCD - parallelogram; Find BD. 1) 20 2) 15 3) 40 4) 10
	A plane α parallel to BC is drawn through the vertex A of triangle ABC. The distance from the point of intersection of the medians of triangle ABC to this plane is 4. How far is BC from the plane? 1) 8 2) 6 3) 12 4) 14
	The point P is removed from all sides of the rhombus at a distance equal to, and is located at a distance equal to 2 from its plane. What is the side of the rhombus if its angle is 30 °? Answer: ____
	In the figure Find the angle between the MC and the AMB plane. 1) 30 0 2) 60 0 3) 90 0 4) 45 0
	Select faithful statements: 1) The angle between a straight line and a plane can be no more than 90 0 . 2) Two planes perpendicular to one straight line intersect. 3) The length of the perpendicular is greater than the length of the oblique drawn from the same point. 4) The diagonal of a rectangular parallelepiped is greater than any of the edges. Answer: ______
	Segment AB rests with ends A and B on the faces of a right dihedral angle. The distances from points A and B to the edge are 2, and the length of the segment AB is 4. Find the length of the projection of this segment onto the edge.
	In the tetrahedron DABC, the base of ABC is a regular triangle. Vertex D is projected into its center O. Find the angle between plane ADO and face DCB. 1) 30 0 2) 60 0 3) 90 0 4) 45 0
	Triangle AMB and rectangle ABCD are arranged so that their planes are mutually perpendicular. Find the MAD angle. 1) 90 0 2) 60 0 3) 30 0 4) 45 0

Test 1

Option 1
Option 2

Test 2

Option 1
Option 2

Test 3

Option 1
Option 2

“Perpendicular lines in space.

Perpendicularity of a line and a plane"

Option 1

Level A

1. Which statement is true?

1) If one of the two lines is perpendicular to the third line, then the other line is also perpendicular to this line.

2) If two lines are perpendicular to a third line, then they are parallel.

3) If two lines are perpendicular to a plane, then they are parallel.

2. ABCD- rectangle, BM ┴ (ABC) . Then it is not true that...

1) BM┴ AC;

2) AM┴ AD;

3) MD┴ DC.

3. Direct m perpendicular to straight lines a and b lying in the plane α, but m not perpendicular to the plane α. Then the lines a and b…

1) parallel;

2) intersect;

3) interbreed.

4. The plane α passes through the vertex A of the rhombus ABCD perpendicular to the diagonal AC. Then the diagonal BD...

1) perpendicular to the plane α;

2) parallel to the plane α;

3) lies in the plane α.

5. a ║ α , b┴α. Then the lines a and b can not be …

1) crossing;

2) perpendicular;

3) parallel.

6. ABCD- parallelogram, BDα, AC┴α. Then ABCD can't be …

1) rectangle;

2) square;

3) rhombus.

1) radii; 2) diameters; 3) chords.

8. Which statement is true:

1) A straight line and a plane not passing through it, perpendicular to another plane, are parallel to each other.

2) A plane and perpendicular to a given plane is also perpendicular to a line parallel to a given plane.

3) A plane perpendicular to a given line is also perpendicular to a plane parallel to a given line.

9. AC ┴ (BDM) . Then the segment BM in a triangle ABC is …

1) median;

2) height;

3) a bisector.

Option 1

https://pandia.ru/text/78/082/images/image006_123.gif" width="17" height="16">( a, VM) = …

https://pandia.ru/text/78/082/images/image003_184.gif" width="13" height="13 src="> α , SM = MV, AM= 2.5 cm, AU= 3 cm. Then AB = …

https://pandia.ru/text/78/082/images/image009_91.gif" width="25" height="23 src="> see AU BD= O. FO ┴ (ABC), FO= see distance from point F up to the top of the square is...

https://pandia.ru/text/78/082/images/image013_21.jpg" align="left" width="120" height="102 src=">

5. ABCD- rectangle. bf ┴ (ABC). CF= 20 cm, D.F.= 25 cm. Then the length of the segment CD equals...

https://pandia.ru/text/78/082/images/image015_17.jpg" align="left" width="103" height="99"> lies in the plane α .

5. ABCD- parallelogram, AB https://pandia.ru/text/78/082/images/image016_17.jpg" align="left" width="114" height="113">crossed.

7. Dhttps://pandia.ru/text/78/082/images/image006_123.gif" width="17" height="16 src="> (AB, CD) =600.

8. Which statement is false?

1) Through any point in space there passes a straight line perpendicular to a given plane, and moreover, only one.

2) Through a point that does not lie on a given line, it is possible to construct only one plane perpendicular to the given line.

3) Through a point not lying on a given line, it is possible to construct only one line perpendicular to the given line.

1. Find the angle between the intersecting diagonals of the faces of the cube.

2. Cube A…D 1 find the angle between the lines AD 1 and CB 1 .

3. The diagonal of a rectangular parallelepiped whose base is a square is twice the side of the base. Find the angles between the diagonals of the parallelepiped that lie in one diagonal section.

1) 45 0 and 45 0 .

2) 90 0 and 90 0 .

3) 30 0 and 60 0 .

4) 60 0 and 120 0 .

4. The diagonal of a rectangular parallelepiped whose base is a square is twice the side of the base. Find the angles between the diagonals of the parallelepiped that lie in different diagonal sections.

1) 45 0 and 135 0 .

2) 90 0 and 90 0 .

3) 30 0 and 150 0 .

4) 60 0 and 120 0 .

5. Find the angle between the crossing edges of a regular triangular pyramid.

6. From a point that does not belong to the plane, a perpendicular is lowered onto it and an oblique line is drawn. Find the projection of the oblique if the perpendicular is 12 cm and the oblique is 15 cm.

7. Find the locus of lines perpendicular to a given line and passing through a given point on it.

2) A plane perpendicular to a given line.

3) A plane parallel to a given line.

4) A plane perpendicular to a given line and passing through a given point.

8. Find the locus of points equidistant from two given points.

1) Perpendicular drawn to the middle of the segment connecting these points.

3) A plane perpendicular to a straight line passing through these points.

4) A plane perpendicular to the segment connecting the given points and passing through its midpoint.

9. From a given point, a perpendicular and an oblique line are drawn to the plane. Knowing that their difference is 25 cm, and the distance between their midpoints is 32.5 cm, find the slope.

10. The ends of the segment are at a distance of 26 cm and 37 cm from this plane. Its orthogonal projection onto the plane is 6 dm. Find a cut.

11. One of the legs of a right-angled isosceles triangle lies in a plane, and the other is inclined to it at an angle of 45 0. Find the angle between the hypotenuse of this triangle and the given plane.

12. Find the angle of inclination of the segment to the plane if its orthogonal projection onto this plane is two times less than the segment itself.

13. Find the locus of points equidistant from all points on the circle.

1) The center of the circle.

2) Circle.

3) A plane perpendicular to the plane of the circle and passing through its center.

14. Find the locus of points equidistant from all sides of the rhombus.

1) Perpendicular drawn to the plane of the rhombus and passing through its top.

2) A plane perpendicular to the plane of the rhombus and passing through its diagonal.

3) Perpendicular drawn to the plane of the rhombus and passing through the point of intersection of its diagonals.

4) A circle inscribed in a rhombus.

15. Find the height of a regular triangular pyramid if the side of its base is a, side edge b.

3) .

16. Find the dihedral angle j between the side faces of a regular quadrangular pyramid, all edges of which are equal to 1.

17. Point A is from one of the two perpendicular planes at a distance of 4 cm, and from the other at 16 cm. Find the distance from the point A to the line of intersection of the planes.

18. Find the dihedral angle at the base of a regular quadrangular pyramid if its height is 2 cm and the side of the base is 4 cm.

19. Point B, remote from the edge of the dihedral angle at a distance a, is the same distance from each of its faces. Find this distance if the dihedral angle is j.

1) a sinj.

2) a cosj.

3) a sin.

4) a cos.

20. Point E belongs to plane a, point F belongs to plane b. The planes are perpendicular. Orthogonal projections of a segment EF, equal to 10 cm, on the plane a and b, respectively, are 8 cm and 7.5 cm. Find the projection of the segment EF to the line of intersection of planes a and a.

ANSWERS

Job number	Test number
	4)	3)	3)	4)	4)	2)	1)
	4)	3)	4)	3)	3)	1)	2)
	2)	4)	2)	3)	4)	1)	4)
	4)	1)	4)	3)	2)	3)	3)
	2)	1)	4)	3)	3)	4)	3)
	2)	2)	2)	2)	3)	4)	3)
	4)	3)	4)	2)	1)	4)	4)
	4)	2)	4)	2)	2)	3)	2)
	3)	3)	3)	1)	4)	3)	3)
	1)	4)	1)	4)	3)	3)	4)
	3)	1)	2)	2)	2)	3)	3)
	2)	2)	3)	3)	1)	2)	1)
	2)	3)	4)	4)	4)	4)	3)
	4)	4)	3)	3)	2)	3)	4)
	3)	4)	3)	2)	1)	2)	4)
	3)	2)	2)	2)	4)	3)	3)
	3)	4)	4)	2)	2)	2)	4)
	4)	3)	2)	4)	3)	2)	2)
	2)	4)	3)	1)	3)	2)	2)
	1)	2)	1)	4)	2)	3)	4)