Г 10 3 perpendicularity in space. Test “Perpendicular lines in space

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

rice. 37	Perpendicular lines can intersect and can be skew. Lemma. If one of two parallel lines is perpendicular to the third line, then the other line is perpendicular to this line. Definition. A line is called perpendicular to a plane if it is perpendicular to any line lying in the plane. They also say that the plane is perpendicular to line a.
rice. 38	If line a is perpendicular to the plane, then it obviously intersects this plane. In fact, if the line a did not intersect the plane, then it would lie in this plane or would be parallel to it. But in both cases there would be lines in the plane that are not perpendicular to line a, for example, lines parallel to it, which is impossible. This means that straight line a intersects the plane.

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

A sign of perpendicularity of a line and a plane.

Notes.

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

Problems and tests on the topic "Topic 5. "Perpendicularity of a line and a plane."

• Perpendicularity of a line and a plane
• Dihedral angle. Perpendicularity of planes - Perpendicularity of lines and planes, grade 10

  Lessons: 1 Assignments: 10 Tests: 1

• Perpendicular and oblique. Angle between a straight line and a plane - Perpendicularity of lines and planes, grade 10

  Lessons: 2 Assignments: 10 Tests: 1

• Parallelism of straight lines, line and plane

  Lessons: 1 Assignments: 9 Tests: 1

• Parallelism of planes - Parallelism of lines and planes, grade 10

  Lessons: 1 Assignments: 8 Tests: 1

The material on the topic summarizes and systematizes the information you know from planimetry about the perpendicularity of straight lines. It is advisable to combine the study of theorems on the relationship between parallelism and perpendicularity of straight lines and planes in space, as well as material on the perpendicular and inclined, with a systematic repetition of the corresponding material from planimetry.

Solutions to almost all calculation problems come down to the application of the Pythagorean theorem and its consequences. In many problems, the possibility of using the Pythagorean theorem or its corollaries is justified by the theorem of three perpendiculars or the properties of parallelism and perpendicularity of planes.

“Perpendicular lines in space.

Perpendicularity of a straight line and a plane"

Option 1

Level A

1. Which statement is true?

1) If one of two lines is perpendicular to the third line, then the other line is 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, B.M. ┴ (ABC) . Then it is not true that...

1) B.M.┴ A.C.;

2) A.M.┴ AD;

3) M.D.┴ DC.

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

1) parallel;

2) intersect;

3) interbreed.

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

1) perpendicular to plane α;

2) parallel to the plane α;

3) lies in the α plane.

5. a ║ α , b┴ α. Then straight a And b can not be …

1) interbreeding;

2) perpendicular;

3) parallel.

6. ABCD– parallelogram, BDα, A.C.┴ 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. A.C. ┴ (BDM) . Then the segment B.M. in a triangle ABC is …

1) median;

2) height;

3) 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, AC= 3 cm. Then AB = …

https://pandia.ru/text/78/082/images/image009_91.gif" width="25" height="23 src=">cm. AC BD= O. F.O. ┴ (ABC), F.O.= cm Distance from point F 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. B.F. ┴ (ABC). CF= 20 cm, DF= 25 cm. Then the length of the segment CD equal...

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

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

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 only one.

2) Through a point not lying on a given line, only one plane can be constructed perpendicular to a given line.

3) Through a point not lying on a given line, you can construct only one line perpendicular to the given line.

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 assignments in geometry for the section “Lines and planes in space”, 1st year of secondary vocational education”

Section No. 3. Straight lines and planes in space

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, straight lines and planes.

Perpendicular and oblique.

The angle between a straight line and a plane. Dihedral angle. Perpendicularity of planes.

Axioms of stereometry

Option 1

	1) ABC 2) DBC 3) DAB 4) DAC
	What plane does point K belong to him? 1) ABC and ABD
	Select faithful sayings: 1) Any three points lie in the same plane. 2) If the center of a circle and its point lie in a plane, then the entire circle lies in this plane. 3) Only one plane passes through three points lying on a straight line. 4) A plane passes through two intersecting lines, and only one. Answer: ______
	Select unfaithful sayings: 1) If three straight lines have a common point, then they lie in the same plane. 3) Two planes can have only two common points. 4) Three straight lines intersecting in pairs at different points lie in the same plane. Answer: ______
	Name the straight line along which 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
	Direct lines AB and CD intersect. A plane is drawn through line AB. Name the line of intersection of this plane with the BCD plane. 1) AC 2) AB 3) BC 4) ВD
	Direct lines AB and CD intersect. A plane is drawn through points B and D. Name the line of intersection of this plane with the ACD plane. 1) AC 2) AB 3) BC 4) ВD

Option 2

	Point P lies on line MN. Name the plane to which point P belongs. 1) ABC 2) DBC 3) DAB 4) DAC
	Which planes does point F belong to? 1) ABC and ACD
	Select faithful sayings: 1) Any four points lie in the same plane. 2) Only one plane passes through a straight line and a point not lying on it. 3) If three points of a circle lie in a plane, then the entire circle lies in this plane. 4) Two planes can have only one common point. Answer: ______
	Select unfaithful sayings: 1) Two circles having a common center lie in the same plane. 3) The three vertices of the triangle belong to the same plane. 4) A plane passes through two parallel lines, and 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 planes ABC and C 1 CB intersect. 1) BC 2) B 1 C 1 3) A 1 B 4) B 1 B
	Direct lines AB and CD intersect. A plane is drawn through straight line CD. Name the line of intersection of this plane with plane ABC. 1) CD 2) AD 3) BC 4) ВD
	Direct 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) ВD

Option 1

	Points M, P, K are the midpoints of edges DA, DB, DC of the DABC tetrahedron. Name a line parallel to the FBC plane. 1) MR 2) RK 3) MK 4) MK and RK
	ABCDA 1 B 1 C 1 D 1 is a rectangular parallelepiped. Which line 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. To which plane is the straight line RK parallel? 1) DAB 2) DBC 3) DAC 4) ABC
	Select faithful sayings: 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 also parallel to it or lies in this plane. 3) There is a line that lies in the plane and is parallel to the line intersecting the given plane. 4) Crossing lines do not have common points. Answer: ______
	1) a || n 2) a || b 3) b || c 4) a || c
	faithful sayings: 1) Straight CD and MN crossed. 2) Straight lines AB and MN lie in the same plane. 3) Lines CD and MN intersect. 4) Direct AB and CD crossing. Answer: ______
	1) a And b – intersecting lines 2) a And b – parallel lines 3) a And b – crossing lines
	Determine the relative position of the lines. 1) a And b – intersecting lines 2) a And b – parallel lines 3) a And b – crossing lines
	Triangles ABC and ABF are arranged so that straight lines AB and FK intersect. How are straight lines AK and BF located?
	In the tetrahedron DABC AB = BC = AC = 20; DA = DB = DC = 40. Through the middle of edge AC is 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 edges DA, DB, DC of the DABC tetrahedron. Name the line parallel to the plane FAB. 1) MR 2) RK 3) MK 4) MK and RK
	ABCDA 1 B 1 C 1 D 1 is a rectangular parallelepiped. Which line is parallel to plane A 1 AD? 1) A 2) b 3) p 4) m
	In the tetrahedron DABC AM = MD, AN = NB. To which plane is the straight line MN parallel? 1) DAB 2) DBC 3) DAC 4) ABC
	Select faithful sayings: 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 whose edges are all sharp. Answer: ______
	Points A, B, C and D are the midpoints of the edges of the 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 sayings: 1) Lines CD and MN intersect. 2) Straight AB and MN crossed 3) Straight lines AB and CD are parallel. 4) Straight 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 – crossing 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 – crossing lines
	Two isosceles triangles ABC and ABD with a common base AB are located so that point C does not lie in the plane ABD. Determine the relative positions of the lines containing the medians of the triangles drawn to the sides BC and ВD. 1) they are parallel 2) they cross 3) they intersect
	In the tetrahedron DABC AB = BC = AC = 10; DA = DB = DC = 20. Through the middle of the edge BC there is a plane parallel to AC and ВD. Find the perimeter of the section. Answer: ____

Option 1

	A plane is drawn through side AB of triangle ABC perpendicular to side BC. Determine the type of triangle relative 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
	Through vertex A of triangle ABC a plane α is drawn parallel to BC. The distance from BC to plane α is 12. Find the distance from the point of intersection of the medians of triangle ABC to this plane. 1) 8 2) 6 3) 12 4) 18
	The height of the rhombus is 12. Point M is equidistant from all sides of the rhombus and is located at a distance of 8 from its plane. What is the distance of point M to the sides of the rhombus? Answer: ____
	Select faithful sayings: 2) Two lines perpendicular to the same plane are parallel. 3) The length of the perpendicular is less than the length of the inclined line drawn from the same point. 4) Two intersecting lines can be perpendicular to the same plane. Answer: ______
	The segment AB rests with ends A and B on the edges 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 intersects BC at point E; Find it.
	Rectangle ABCD and parallelogram BEMC are located so that their planes are mutually perpendicular. Find the angle MCD.

Perpendicularity of lines and planes

Option 2

	Through side AD of parallelogram ABCD, a plane is drawn perpendicular to side DC. Determine the type of triangle ABC. 1) acute-angled 2) rectangular 3) obtuse-angled
	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
	Through vertex A of triangle ABC a plane α is drawn parallel to BC. The distance from the point of intersection of the medians of triangle ABC to this plane is 4. At what distance from the plane is BC? 1) 8 2) 6 3) 12 4) 14
	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 MC and plane AMB. 1) 30 0 2) 60 0 3) 90 0 4) 45 0
	Select faithful sayings: 1) The angle between the straight line and the plane can be no more than 90 0. 2) Two planes perpendicular to one line intersect. 3) The length of the perpendicular is greater than the length of the inclined line drawn from the same point. 4) The diagonal of a rectangular parallelepiped is greater than any of the edges. Answer: ______
	The segment AB rests with ends A and B on the edges 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 ABC is a regular triangle. The vertex D is projected to its center O. Find the angle between the plane ADO and the 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 angle MAD. 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

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

2. In a cube A…D 1 find the angle between the lines AD 1 and C.B. 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 the same 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 not belonging to the plane, a perpendicular is dropped onto it and an inclined one is drawn. Find the oblique projection 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) A perpendicular drawn to the middle of a segment connecting these points.

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

4) A plane perpendicular to the segment connecting these points and passing through its middle.

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

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

11. One of the legs of a right 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 half the size of the segment itself.

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

1) 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) A perpendicular drawn to the plane of the rhombus and passing through its vertex.

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

3) A 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 equal to a, side rib b.

3) .

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

17. Point A is located at a distance of 4 cm from one of two perpendicular planes, and at a distance of 16 cm from the other. 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, removed 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 E.F., equal to 10 cm, on plane a and b are respectively 8 cm and 7.5 cm. Find the projection of the segment E.F. to the line of intersection of planes a and a.

ANSWERS

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

Title: Geometry. 10-11 grade. Tests

The manual contains tests on the main topics of the geometry course for grades 10-11 in two versions - 8 tests for grade 10 and 9 tests for grade 11.
The teacher can use the proposed tests to monitor students' knowledge before conducting a test or as a test. Students can use tests in self-preparation for final exams, as well as for entrance exams to universities.

This book presents testing tests in geometry for grades 10-11. It is a continuation of a similar book on geometry for grades 7-9. Tests are given in two versions - 8 tests for grade 10 and 9 tests for grade 11.
It is advisable to conduct tests once a month as testing before tests or replacing them. Given the complexity of individual tasks, two lessons should be allocated to complete the full test. However, the teacher can divide the test into 2 parts (4 tasks each) and give it in two different lessons on different days. In this case, the teacher must take into account the fact that the tasks are not arranged in order of increasing difficulty (i.e., for example, task 3 may be more difficult than task 5); this was done deliberately so that students solve not only easy problems, but also tried to solve more complex ones. But the teacher, having reviewed the tasks of a separate test, can himself vary the number and complexity of the tasks.
Taking into account the unique nature of conducting verification tests, when the given answers to some extent facilitate the solution of the problem, the teacher can conduct an analysis of the work in the next lesson, placing emphasis on the theoretical justification for solving problems, conducting the necessary evidence in order to identify the logical validity of the student’s choice of answer.
The sequence of material is given in accordance with the textbook on geometry for grades 7-11 by A.V. Pogorelov. However, teachers working with other teaching aids, having made the necessary adjustments, can also use them in their work.

Content
Preface
Grade 10
Test 1. Axioms of stereometry. Corollaries from the axioms
Test 2. Parallelism in space
Test 3. Perpendicularity in space
Test 4. Parallelism and perpendicularity in space
Test 5. Coordinates in space
Test 6. Angles between straight lines and planes
Test 7. Vectors
Test 8. Final
Grade 11
Test 1. Dihedral and linear angles. Polyhedral angles
Test 2. Parallelepiped and prism
Test 3. Pyramid. Truncated pyramid
Test 4. Cylinder. Cone. Ball
Test 5. Volumes of polyhedra
Test 6. Volumes of bodies of revolution
Test 7. Combinations of figures
Test 8. Final - 1
Test 9. Final - 2
Answers

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