Tasks and exercises on ready-made drawings, grades 10-11, Geometry, Rabinovich E. M., 2006.

Table of contents
Preface.
Repetition of the course of planimetry.
Table 1. Solution of triangles.
Table 2. Area of ​​a triangle.
Table 3. Area of ​​a quadrilateral.
Table 4. Area of ​​a quadrilateral. Stereometry. Grade 10.
Table 10.1. Axioms of stereometry and their simplest consequences.
Table 10.2. Axioms of stereometry and their simplest consequences.
Table 10.3. Parallelism of lines in space. Crossing straight lines.
Table 10.4. Parallelism of lines and planes.
Table 10.5. A sign of parallel planes.
Table 10.6. Properties of parallel planes.
Table 10.7. Image of spatial figures on a plane
Table 10.8. Image of spatial figures on a plane
Table 10.9. Perpendicularity of a line and a plane.
Table 10.10. Perpendicularity of a line and a plane.
Table 10.11. Perpendicular and oblique.
Table 10.12. Perpendicular and oblique.
Table 10.13. Theorem on three perpendiculars.
Table 10.14. Theorem on three perpendiculars.
Table 10.15. Theorem on three perpendiculars.
Table 10.16. Plane perpendicularity.
Table 10.17. Plane perpendicularity.
Table 10.18. Distance between intersecting lines.
Table 10.19. Cartesian coordinates in space.
Table 10.20. Angle between intersecting lines.
Table 10.21. The angle between a line and a plane.
Table 10.22. Angle between planes.
Table 10.23. Area of ​​an orthogonal projection of a polygon
Table 10.24. Vectors in space. Stereometry. Grade 11.
Table 11.1. Dihedral angle. triangular angle.
Table 11.2. direct prism.
Table 11.3. correct prism.
Table 11.4. correct prism.
Table 11.5. tilted prism.
Table 11.6. Parallelepiped.
Table 11.7. Construction of sections of a prism.
Table 11.8. Correct pyramid.
Table 11.9. Pyramid.
Table 11.10. Pyramid.
Table 11.11. Pyramid. Truncated pyramid.
Table 11.12. Construction of a section of a pyramid.
Table 11.13. Cylinder.
Table 11.14. Cone.
Table 11.15. Kohuc. Truncated kohuc.
Table 11.16. Ball.
Table 11.17. Inscribed and circumscribed sphere.
Table 11.18. The volume of the parallelepiped.
Table 11.19. The volume of the prism.
Table 11.20. volume of the pyramid.
Table 11.21. volume of the pyramid.
Table 11.22. volume of the pyramid. The volume of a truncated pyramid.
Table 11.23. The volume and area of ​​the lateral surface of the cylinder.
Table 11.24. The volume and area of ​​the lateral surface of the cone.
Table 11.25. Cone volume. The volume of a truncated cone. The area of ​​the lateral surface of the cone. The area of ​​the lateral surface of a truncated cone.
Table 11.26. The volume of the ball. The surface area of ​​a sphere. Answers, directions, solutions

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Title: Geometry. Tasks and exercises on ready-made drawings. 10-11 class.

A mathematics teacher working in high school knows well how difficult it is to teach students to make visual and correct drawings for stereometric problems.
Due to the lack of spatial imagination, the stereometric task, for which you need to make a drawing yourself, often becomes overwhelming for the student.
That is why the use of ready-made drawings for stereometric tasks significantly increases the amount of material considered in the lesson, increases its effectiveness.
The proposed manual is an additional collection of problems in geometry for students in grades 10-11 of a general education school and is focused on the textbook by A.V. Pogorelov "Geometry 7-11. It is a continuation of a similar manual for students in grades 7-9.
The manual is compiled in the form of tables and contains more than 350 tasks. The tasks of each table correspond to a specific topic of the school geometry course for grades 10-11 and are located inside the table in order of increasing complexity.

Table of contents
Foreword
Repetition of the course of planimetry
Table 1. Solution of triangles
Table 2. Area of ​​a triangle
Table 3. Area of ​​a quadrilateral
Table 4
Stereometry. Grade 10
Table 10.1 Axioms of stereometry and their simplest consequences
Table 10.2. Axioms of stereometry and their simplest consequences
Table 10.3. Parallelism of lines in space. Crossed lines
Table 10.4. Parallelism of lines and planes
Table 10.5. Sign of parallel planes
Table 10.6. Properties of Parallel Planes
Table 10.7. Image of spatial figures on a plane
Table 10.8. Image of spatial figures on a plane
Table 10.9. Perpendicularity of a line and a plane
Table 10.10. Perpendicularity of a line and a plane
Table 10.11. Perpendicular and oblique
Table 10.12. Perpendicular and oblique
Table 10.13. Three perpendiculars theorem
Table 10.14. Three perpendiculars theorem
Table 10.15. Three perpendiculars theorem
Table 10.16. Plane perpendicularity
Table 10.17. Plane perpendicularity
Table 10.18. Distance between intersecting lines
Table 10.19. Cartesian coordinates in space
Table 10.20. Angle between skew lines
Table 10.21. Angle between line and plane
Table 10.22. Angle between planes
Table 10.23. Area of ​​an orthogonal projection of a polygon
Table 10.24. Vectors in space
Stereometry. Grade 11
Table 11.1. Dihedral angle. trihedral angle
Table 11.2. straight prism
Table 11.3. Correct prism
Table 11.4. Correct prism
Table 11.5. tilted prism
Table 11.6. Parallelepiped
Table 11.7. Construction of sections of a prism
Table 11.8. Correct pyramid
Table 11.9. Pyramid
Table 11.10. Pyramid
Table 11.11. Pyramid. Truncated pyramid
Table 11.12. Construction of sections of the pyramid
Table 11.13. Cylinder
Table 11.14. Cone
Table 11.15. Cone. Frustum
Table 11.16. Ball
Table 11.17. Inscribed and circumscribed sphere
Table 11.18. Volume of the box
Table 11.19. Prism volume
Table 11.20. Pyramid Volume
Table 11.21. Pyramid Volume
Table 11.22. volume of the pyramid. Volume of a truncated pyramid
Table 11.23. Volume and area of ​​the lateral surface of the cylinder
Table 11.24. Volume and area of ​​the lateral surface of the cone
Table 11.25. Cone volume. The volume of a truncated cone. The area of ​​the lateral surface of the cone. Lateral surface area of ​​a truncated cone
Table 11.26. The volume of the ball. Ball surface area
Answers, directions, solutions

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Block of lessons on the topic: “Pyramid”

(9 lessons, of which 1 lesson is a test)

Goals:

1. Introduce the concept of a pyramid, its elements, the concept of a regular and truncated pyramid, consider sections of a pyramid.
2. To teach how to solve problems for finding the elements of a pyramid, the area of ​​\u200b\u200bthe side and full surfaces of the pyramid, tasks for constructing sections of the pyramid.
3. Develop spatial representation, creative and logical thinking in the course of solving problems.
4. Apply level differentiation, teach children to independently choose the level of preparation of their knowledge.

TOPIC PLAN

1. Lesson-lecture with reference notes.
2. Lesson for solving learning problems for the construction of sections.
3. Lesson test to test knowledge of theory. Solution of basic problems.
4. Solving basic problems and problems of an advanced level.
5. Solving problems at an advanced level (at the end of the lesson, independent work on homework).
6. Lesson-consultation (solving problems from independent work that caused difficulties)
7. Lesson-seminar on the topic.
8. Regular polyhedra.
9. Test.

First lesson .

Lecture with reference notes.

Objectives: to introduce the concepts of a pyramid, its elements, a truncated and regular pyramid, to consider sections of a pyramid, to derive a formula for finding the lateral surface area of ​​a regular pyramid.

Lecture plan:

1. Definition of a pyramid, its elements.
2. Image of a pyramid and its sections.
3. Truncated pyramid.
4. Correct pyramid. The position of the base of the height of a regular pyramid.
5. The formula for the lateral surface area of ​​a regular pyramid.

Homework: to teach a lecture, paragraphs 176-179 from A.V. Pogorelov.

Reporting topics of creative tasks.

1. First information about parallelepipeds and their properties.

2. The meaning of the pyramids from a historical and mathematical point of view.

3. Platonic solids and their properties.

4. Euler. Theorem on the number of faces, vertices and edges of a polyhedron.

5. Archimedes and his "bodies".

6. Geometry in the kitchen.

7. Comfort in the room.

8. History of geometry: from the ancient world to the present day.

9. History of measurement of areas and volumes.

10. Polyhedra and bodies of revolution in architecture.

11. Polyhedra in the world of chemistry.

Lesson two.

The solution of training problems for the construction of sections of the pyramid.

Objectives: consolidation of the studied concepts, development of spatial representation.

During the classes:

1. Frontal conversation on the last lecture.
2. Solving problems on the construction of sections.

D/m A.P. Ershov. C-8, Option A 1 #1,2,3, Option B 1 #2,3.

1. Independently with verification Option A 2 #2, Option B 2 #1.
2. Homework: Option A 2 №3, Option B 2 No. 2,3. Repeat n. 67,160-162.

Lesson three.

Checking knowledge of the theory and solving basic problems.

Objectives: to control students' knowledge on the topic and their application in solving elementary problems.

During the classes:

1. Theory report.

1. Knowledge update. Repetition of the rules for solving right triangles, the concepts of angles between planes, between a line and a plane.
2. Solving problems on finding elements of a regular quadrangular pyramid.

Collection of tasks. No. 1-4 oral decision on the finished drawing.

№5 table 1. (№1,2,4,5) table 2. (№1,3)

Homework: Collection of tasks. №5 table 1. (№3,6,7) table 2. (№2,4)

Repeat paragraphs 116, 127.

Lesson four.

Solving support problems for finding elements of a regular triangular pyramid, problems for applying the properties of pyramids with equal side edges, equal apothems.

Objectives: to teach to apply knowledge of theory in solving elementary problems.

During the classes:

1. Checking homework.
2. Knowledge update. Repetition of the properties of a regular triangle, area formulas, radii of the inscribed and circumscribed circles.
3. Problem solving. Collection of tasks. №10 table 3. (1.3) table 4. (1.2)

No. 11. table 5. (1,3)

1. Consider the properties of pyramids having equal side edges, equal apothems. Group work. Collection of tasks. No. 6, 7,8,9.
2. Homework: Collection of tasks. №10 table 3. (2.4) table 4. (3.4)

No. 11. table 5. (2.5) Repeat paragraphs 143-148.

Lesson five.

Problem solving.

Objectives: to learn to apply in solving problems the properties of perpendicularity of planes, perpendicularity of a straight line and a plane, the theorem of three perpendiculars.

During the classes:

1. Checking homework. Knowledge update. Repetition of the properties of perpendicularity in space.
2. Problem solving. Collection of tasks. No. 13, 14, 15.
3. Checking independent work.

D/M E.M. Rabinovich Tables 11.8;11.9,11.10 No. 1,2,3,4, Students independently choose the level of difficulty of the tasks.

1. Homework: Collection of tasks. No. 12, 16.

Lesson six.

Consultation on solving problems from independent work. Solving problems on the application of the properties of a regular truncated pyramid.

Objectives: correction of knowledge, consolidation of concepts related to the correct truncated pyramid.

During the classes:

1. Consultation on solving problems from independent and homework. Those students who do not have questions complete tasks from the d / m E.M. Rabinovich Table 11.12.
2. Knowledge update. Repetition of the properties of a regular truncated pyramid, the properties of a trapezoid.
3. Problem solving. A.V. Pogorelov. No. 70,72,73.
4. Discussion of the topic of the seminar, distribution of tasks into groups.
5. Homework: #71, prepare for the seminar.

Lesson seven.

Seminar on the topic: "Pyramid"

Objectives: to promote the development of students' ability to use additional literature, the ability to put forward hypotheses and prove them, to form teamwork skills.

3-4 days are allotted for preparation for the seminar. The class is divided into 5 groups, each of which receives one of the 5 tasks listed in the workshop plan.

Each group, in preparation for the seminar, works through the relevant sections of the textbook and lectures, and also uses additional literature, and receives advice from the teacher.

Seminar plan:

1. Report on the topic: “A polyhedron is a body or a surface. Types of polyhedrons". (1 group)

2. Regular polyhedra. Development of a regular polyhedron. (Group 2)

3. Solving problems for the study of the position of the base of the height of the pyramid. (groups 3,4 and 5) A.P. Ershov. C-10, Option A 2 No. 1, C-11, Option B 2 No. 1,2.

4. Homework: D/m A.P. Ershov. C-10, Option A 1 No. 1, C-11, Option B 1 No. 1.2.

Lesson eight.

Lecture-presentation on the topic: "Regular polyhedra".

Lecture plan

1. "Magnificent Five" (presentation about five regular polyhedra) Students make notes in their notebooks during the lecture.
2. Presentation "Flowers from the Geometry Garden"

1. Definition of a regular convex polyhedron.

2. Platonic solids, their types.

3. Euler's formula for convex polyhedra.

4. Formulas for calculating the volume and surface area of ​​regular polyhedra.

5. The use of the form of regular polyhedra by nature and man.

6. Star polyhedra, their types.

7. Archimedean solids, their types.

Homework: make models of polyhedra from various materials, prepare for the test.

Lesson nine.

Test work on the topic: "Pyramid"

Objectives: to test the knowledge of children on this topic.

Collection of tasks on the topic: "Pyramid".

1. In a regular quadrangular pyramid, the height makes an angle of 37 ° with the side face. Find the angle between the apothems of opposite side faces.

2. The lateral edge of a regular pyramid is twice its height. Determine the angle of inclination of the side edge to the plane of the base.

3. Find the value of the dihedral angle at the base of a regular quadrangular pyramid, if the height of the pyramid is half the side of the base.

4. The height of a regular quadrangular pyramid is equal to half the diagonal of the base. What is the angle of inclination of the lateral edge to the plane of the base?

Table 1

table 2

6. The base of the KAVSD pyramid is a rectangle with sides 6 cm and 8 cm. Each side edge of the pyramid is 13 cm. Find the height of the pyramid and the area of ​​the lateral surface.

7. The base of the pyramid is a rhombus with a side of 8 cm and an angle of 30 0 . The side faces of the pyramid form angles of 60 with the plane of the base 0 . Find the surface area of ​​the pyramid and the height.

8. The base of the pyramid is a right triangle with a hypotenuse of 10 cm. The side edges of the pyramid form angles of 45 with the base plane. 0 . Find the height of the pyramid.

9. The base of the pyramid is a triangle with sides 13cm, 14cm, 15cm. The heights of the side faces of the pyramid are 5 cm. Find the height of the pyramid.

Table 3

Table 4

12. The side edges of the pyramid are equal to the hypotenuse of the right triangle at the base and equal to 12 cm. Calculate the height of the pyramid.

13 . In a regular quadrangular pyramid, the lateral edge is 20 cm, it makes an angle of 45 ° with the base. Determine the distance from the center of the base to the side edge.

14 . The base of the pyramid is square ABSD. The height of the pyramid passes through the vertex B. Prove that all the side faces of the pyramid are right triangles.

15. The base of the pyramid MAVSD is a rectangle ABSD with sides AB=5cm, BC=15cm. The faces of MAB and MVS are perpendicular to the base. Find the angles of inclination of the MSD and MAD faces to the base plane if the height of the pyramid is 5 cm.

16 . The base of the pyramid DAVS is a right triangle ABC, AB=6cm, angle A=30 0 , the angle C is a right. The faces of DAS and DSW are perpendicular to the plane of the base DS=10cm. Find the area of ​​the DAW face and the angle between the planes DSA and DSW.

Literature

1. R.B. Reichmist. Mathematical task book for secondary school students and university applicants. - M .: "Moscow Lyceum", 2001.
2. L.O. Denishcheva, T.F. Mikheev. Learning to solve problems. - M .: "Intellect Center", 2000.
3. T.L. Afanasiev, L.A. Tapilina. "Geometry grade 11" (lesson plans) - Volgograd .: "Teacher", 1999.
4. T.N. Mishchenko Workbook. Grade 11. To the textbook L.S. Atanasyan and others. Geometry grade 10-11.

Geometry. Tasks and exercises on ready-made drawings. 10-11 grades. Rabinovich E.M.

M.: 2014. - 80 p.

The manual is compiled in the form of tables and contains more than 350 tasks. The tasks of each table correspond to a specific topic of the school geometry course for grades 10-11 and are located inside the table in order of increasing complexity.

A mathematics teacher working in high school knows well how difficult it is to teach students to make visual and correct drawings for stereometric problems.

Due to the lack of spatial imagination, the stereometric task, for which you need to make a drawing yourself, often becomes overwhelming for the student.

That is why the use of ready-made drawings for stereometric tasks significantly increases the amount of material considered in the lesson, increases its effectiveness.

The proposed manual is an additional collection of problems in geometry for students in grades 10-11 of a general education school and is focused on the textbook by A.V. Pogorelov "Geometry 7-11". It is a continuation of a similar manual for students in grades 7-9.

Format: pdf(2014, 80s.)

Size: 1.2 MB

Watch, download:drive.google ; Rghost

Format: djvu(2006, 80s.)

Size: 1.3 MB

Download: drive.google

Table of contents
Preface 3
Repetition of the course of planimetry 5
Table 1. Solution of triangles 5
Table 2. Area of ​​triangle 6
Table 3. Area of ​​quadrilateral 7
Table 4. Area of ​​quadrilateral 8
Stereometry. 10 grade 9
Table 10.1. Axioms of stereometry and their simplest consequences... 9
Table 10.2. Axioms of stereometry and their simplest consequences. 10
Table 10.3. Parallelism of lines in space. Crossed lines 11
Table 10.4. Parallelism of lines and planes 12
Table 10.5. Sign of parallel planes 13
Table 10.6. Properties of parallel planes 14
Table 10.7. Image of spatial figures on a plane 15
Table 10.8. Image of spatial figures on a plane 16
Table 10.9. Perpendicularity of line and plane 17
Table 10.10. Perpendicularity of line and plane 18
Table 10.11. Perpendicular and oblique 19
Table 10.12. Perpendicular and oblique 20
Table 10.13. Three perpendiculars theorem 21
Table 10.14. Three perpendiculars theorem 22
Table 10.15. Three perpendiculars theorem 23
Table 10.16. Plane Perpendicularity 24
Table 10.17. Plane Perpendicularity 25
Table 10.18. Distance between crossing lines 26
Table 10.19. Cartesian coordinates in space 27
Table 10.20. Angle between skew lines 28
Table 10.21. Angle between line and plane 29
Table 10.22. Angle between planes 30
Table 10.23. Area of ​​the orthogonal projection of the polygon 31
Table 10.24. Vectors in space 32
Stereometry. 11 class 33
Table 11.1. Dihedral angle. Trihedral angle 33
Table 11.2. Straight prism 34
Table 11.3. Correct prism 35
Table 11.4. Correct prism 36
Table 11.5. Inclined prism 37
Table 11.6. Parallelepiped 38
Table 11.7. Construction of sections of a prism 39
Table 11.8. Correct pyramid 40
Table 11.9. Pyramid 41
Table 11.10. Pyramid 42
Table 11.11. Pyramid. Truncated Pyramid 43
Table 11.12. Sectioning a Pyramid 44
Table 11.13. Cylinder 45
Table 11.14. Cone 46
Table 11.15. Cone. Truncated cone 47
Table 11.16. Ball 48
Table 11.17. Inscribed and circumscribed sphere 49
Table 11.18. The volume of the parallelepiped is 50
Table 11.19. Prism Volume 51
Table 11.20. Pyramid Volume 52
Table 11.21. Pyramid Volume 53
Table 11.22. volume of the pyramid. The volume of the truncated pyramid 54
Table 11.23. The volume and area of ​​the lateral surface of the cylinder..55
Table 11.24. Volume and area of ​​the lateral surface of the cone 56
Table 11.25. Cone volume. The volume of a truncated cone. The area of ​​the lateral surface of the cone. The area of ​​the lateral surface of the truncated cone 57
Table 11.26. The volume of the ball. Ball surface area 58
Answers, instructions, solutions 59

Geometry. Tasks and exercises on ready-made drawings. 10-11 grades. Rabinovich E.M.

M.: 2014. - 80 p.

The manual is compiled in the form of tables and contains more than 350 tasks. The tasks of each table correspond to a specific topic of the school geometry course for grades 10-11 and are located inside the table in order of increasing complexity.

A mathematics teacher working in high school knows well how difficult it is to teach students to make visual and correct drawings for stereometric problems.

Due to the lack of spatial imagination, the stereometric task, for which you need to make a drawing yourself, often becomes overwhelming for the student.

That is why the use of ready-made drawings for stereometric tasks significantly increases the amount of material considered in the lesson, increases its effectiveness.

The proposed manual is an additional collection of problems in geometry for students in grades 10-11 of a general education school and is focused on the textbook by A.V. Pogorelov "Geometry 7-11". It is a continuation of a similar manual for students in grades 7-9.

Format: pdf(2014, 80s.)

Size: 1.2 MB

Watch, download:drive.google ; Rghost

Format: djvu(2006, 80s.)

Size: 1.3 MB

Download: drive.google

Table of contents
Preface 3
Repetition of the course of planimetry 5
Table 1. Solution of triangles 5
Table 2. Area of ​​triangle 6
Table 3. Area of ​​quadrilateral 7
Table 4. Area of ​​quadrilateral 8
Stereometry. 10 grade 9
Table 10.1. Axioms of stereometry and their simplest consequences... 9
Table 10.2. Axioms of stereometry and their simplest consequences. 10
Table 10.3. Parallelism of lines in space. Crossed lines 11
Table 10.4. Parallelism of lines and planes 12
Table 10.5. Sign of parallel planes 13
Table 10.6. Properties of parallel planes 14
Table 10.7. Image of spatial figures on a plane 15
Table 10.8. Image of spatial figures on a plane 16
Table 10.9. Perpendicularity of line and plane 17
Table 10.10. Perpendicularity of line and plane 18
Table 10.11. Perpendicular and oblique 19
Table 10.12. Perpendicular and oblique 20
Table 10.13. Three perpendiculars theorem 21
Table 10.14. Three perpendiculars theorem 22
Table 10.15. Three perpendiculars theorem 23
Table 10.16. Plane Perpendicularity 24
Table 10.17. Plane Perpendicularity 25
Table 10.18. Distance between crossing lines 26
Table 10.19. Cartesian coordinates in space 27
Table 10.20. Angle between skew lines 28
Table 10.21. Angle between line and plane 29
Table 10.22. Angle between planes 30
Table 10.23. Area of ​​the orthogonal projection of the polygon 31
Table 10.24. Vectors in space 32
Stereometry. 11 class 33
Table 11.1. Dihedral angle. Trihedral angle 33
Table 11.2. Straight prism 34
Table 11.3. Correct prism 35
Table 11.4. Correct prism 36
Table 11.5. Inclined prism 37
Table 11.6. Parallelepiped 38
Table 11.7. Construction of sections of a prism 39
Table 11.8. Correct pyramid 40
Table 11.9. Pyramid 41
Table 11.10. Pyramid 42
Table 11.11. Pyramid. Truncated Pyramid 43
Table 11.12. Sectioning a Pyramid 44
Table 11.13. Cylinder 45
Table 11.14. Cone 46
Table 11.15. Cone. Truncated cone 47
Table 11.16. Ball 48
Table 11.17. Inscribed and circumscribed sphere 49
Table 11.18. The volume of the parallelepiped is 50
Table 11.19. Prism Volume 51
Table 11.20. Pyramid Volume 52
Table 11.21. Pyramid Volume 53
Table 11.22. volume of the pyramid. The volume of the truncated pyramid 54
Table 11.23. The volume and area of ​​the lateral surface of the cylinder..55
Table 11.24. Volume and area of ​​the lateral surface of the cone 56
Table 11.25. Cone volume. The volume of a truncated cone. The area of ​​the lateral surface of the cone. The area of ​​the lateral surface of the truncated cone 57
Table 11.26. The volume of the ball. Ball surface area 58
Answers, instructions, solutions 59