Solving problems using ready-made drawings of a “regular triangular pyramid”. Solving problems using ready-made drawings “regular triangular pyramid” Table 11.5 inclined prism solution

Regular triangular pyramid Solving problems using ready-made drawings MBOU Verkhnyakovskaya Secondary School Mathematic teacher: Martynenko L.N. DABC is a regular pyramid, DO ┴ (ABC),CK ┴ AB AM ┴ BC BN ┴ AC. Task #1: Find DO

• Tips:
• Find DK
• Apply the property of medians of a triangle
• Apply the Pythagorean theorem to find DO

DABC is a regular pyramid, DO ┴ (ABC), CK ┴ AB, AM ┴ BC, BN ┴ AC. Task No. 2: Find P of the base.

• Tips:
• Apply the law of cosines

DABC is a regular pyramid, DO is perpendicular (ABC) Problem No. 3: O1 and O2 – the intersection points of the medians of triangles ABD and BCD, respectively O1O2 = 2. Find S base.

• Tips:
• Consider triangles KDM and DO1O2
• Find KM
• Using the property of the midline of a triangle, find the side of the triangle

DABC is a regular pyramid, DO ┴ (ABC), CK ┴ AB, AM ┴ BC, BN ┴ AC. Task No. 4: Find DO

• Tips:
• Use the property of medians of a triangle
• Apply the Pythagorean theorem to find the height

DABC is a regular pyramid, DO ┴ (ABC), CK ┴ AB, AM ┴ BC, BN ┴ AC. Problem #5: Find the angle DKC

• Tips:
• Apply the bisector property of a triangle

DABC is a regular pyramid, DO ┴ (ABC), CK ┴ AB, AM ┴ BC, BN ┴ AC. Problem #6: Find DO

• Tips:
• Which element needs to be found to calculate DO?
• Use the property of medians of a triangle and ratios in a right triangle

DABC is a regular pyramid, DO ┴ (ABC), CK ┴ AB, AM ┴ BC, BN ┴ AC. Task No. 7: Find the apothem DM.

• Tips:
• Apply the property of medians of a triangle to find OM

DABC is a regular pyramid, DO ┴ (ABC), CK ┴ AB, AM ┴ BC, BN ┴ AC. Problem #8: Find COS

• Tips:
• Use the property of medians of a triangle and ratios in a right triangle

DABC is a regular pyramid, DO ┴ (ABC),CK ┴ AB AM ┴ BC BN ┴ AC. Problem #9: Find COS

• Tips:
• Use the property of medians in a triangle and ratios in a right triangle

DABC is a regular pyramid, DO ┴ (ABC),CK ┴ AB AM ┴ BC BN ┴ AC. Problem #10: Find SPDL

• Tips:
• Find DO

DABC is a regular pyramid, DO ┴ (ABC),CK ┴ AB AM ┴ BC BN ┴ AC. Problem #11: Find SPQL

• Tips:
• Write down the formula for the area of ​​a triangle
• Find PL from the similarity of triangles ABC and APL
• Find QL from the similarity of triangles ADC and AQL
• Find the height of triangle PQL using Pythagorean theorem

DABC is a regular pyramid, DO ┴ (ABC),CK ┴ AB AM ┴ BC BN ┴ AC. Task #12: Find SDKC

• Tips:
• Write down the formula for the area of ​​a triangle
• Find CK
• Use the property of medians of a triangle to find CO
• Find the height of triangle CDK

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

Table of contents
Preface 3
Repetition of planimetry course 5
Table 1. Solving triangles 5
Table 2. Area of ​​triangle 6
Table 3. Area of ​​quadrilateral 7
Table 4. Area of ​​quadrilateral 8
Stereometry. 10th 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. Crossing 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 a line and a plane 17
Table 10.10. Perpendicularity of a straight line and a plane 18
Table 10.11. Perpendicular and oblique 19
Table 10.12. Perpendicular and oblique 20
Table 10.13. Theorem of three perpendiculars 21
Table 10.14. Theorem of three perpendiculars 22
Table 10.15. Theorem of three perpendiculars 23
Table 10.16. Perpendicularity of planes 24
Table 10.17. Perpendicularity of planes 25
Table 10.18. Distance between crossing lines 26
Table 10.19. Cartesian coordinates in space 27
Table 10.20. Angle between crossing lines 28
Table 10.21. Angle between straight line and plane 29
Table 10.22. Angle between planes 30
Table 10.23. Area of ​​orthogonal projection of a polygon 31
Table 10.24. Vectors in space 32
Stereometry. 11th grade 33
Table 11.1. Dihedral angle. Triangular 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. Constructing prism sections 39
Table 11.8. Regular pyramid 40
Table 11.9. Pyramid 41
Table 11.10. Pyramid 42
Table 11.11. Pyramid. Truncated pyramid 43
Table 11.12. Constructing pyramid sections 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 ball 49
Table 11.18. Volume of parallelepiped 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. Volume of a truncated pyramid 54

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

A high school mathematics teacher knows well how difficult it is to teach students to make visual and correct drawings for stereometric problems.
Due to a lack of spatial imagination, a 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 problems significantly increases the volume of material covered in the lesson and increases its effectiveness.
The proposed manual is an additional collection of geometry problems for students in grades 10-11 of a general education school and is focused on the textbook by A.V. Pogorelova "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
Preface
Repeating the planimetry course
Table 1. Solving triangles
Table 2. Area of ​​the triangle
Table 3. Area of ​​the quadrilateral
Table 4. Area of ​​the 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 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 Perpendicular Theorem
Table 10.14. Three Perpendicular Theorem
Table 10.15. Three Perpendicular Theorem
Table 10.16. Perpendicularity of planes
Table 10.17. Perpendicularity of planes
Table 10.18. Distance between crossing lines
Table 10.19. Cartesian coordinates in space
Table 10.20. Angle between intersecting lines
Table 10.21. Angle between a straight line and a plane
Table 10.22. Angle between planes
Table 10.23. Area of ​​orthogonal projection of a polygon
Table 10.24. Vectors in space
Stereometry. Grade 11
Table 11.1. Dihedral angle. Triangular angle
Table 11.2. Straight prism
Table 11.3. Correct prism
Table 11.4. Correct prism
Table 11.5. Oblique prism
Table 11.6. Parallelepiped
Table 11.7. Constructing prism sections
Table 11.8. Correct pyramid
Table 11.9. Pyramid
Table 11.10. Pyramid
Table 11.11. Pyramid. Truncated pyramid
Table 11.12. Constructing pyramid sections
Table 11.13. Cylinder
Table 11.14. Cone
Table 11.15. Cone. Frustum
Table 11.16. Ball
Table 11.17. Inscribed and circumscribed ball
Table 11.18. Volume of a parallelepiped
Table 11.19. Prism volume
Table 11.20. Volume of the pyramid
Table 11.21. Volume of the pyramid
Table 11.22. Volume of the pyramid. Volume of a truncated pyramid
Table 11.23. Volume and lateral surface area of ​​the cylinder
Table 11.24. Volume and lateral surface area of ​​a cone
Table 11.25. Cone volume. Volume of a truncated cone. The area of ​​the lateral surface of the cone. Lateral surface area of ​​a truncated cone
Table 11.26. Volume of the ball. Sphere surface area
Answers, directions, solutions

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Block of lessons on the topic: “Pyramid”

(9 lessons, 1 of which is a test)

Goals:

1. Introduce the concept of a pyramid, its elements, the concept of a regular and truncated pyramid, consider the sections of the pyramid.
2. Learn to solve problems on finding the elements of a pyramid, the area of ​​the lateral and full surfaces of a pyramid, problems on constructing sections of a pyramid.
3. Develop spatial understanding, creative and logical thinking while solving problems.
4. Apply level differentiation, teach children to independently choose the level of preparation of their knowledge.

TOPIC PLAN

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

First lesson .

Lecture with supporting notes.

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

Lecture outline:

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

Homework: study the lecture, pp. 176-179 from the textbook by A.V. Pogorelova.

Reporting topics for creative assignments.

1. First information about parallelepipeds and their properties.

2. The significance of the pyramids from historical and mathematical points 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 measuring areas and volumes.

10. Polyhedra and bodies of rotation in architecture.

11. Polyhedra in the world of chemistry.

Lesson two.

Solving educational problems on constructing sections of a pyramid.

Goals: consolidation of learned concepts, development of spatial representation.

During the classes:

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

D/m A.P. Ershova. S-8, Option A 1 No. 1,2,3, OptionB 1 No. 2,3.

1. Independently with verification Option A 2 No. 2, Option B 2 No. 1.
2. Homework: Option A 2 No. 3, Option B 2 No. 2,3. Repeat paragraphs 67.160-162.

Lesson three.

Testing theory knowledge and solving basic problems.

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

During the classes:

1. Theory test.

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

Collection of problems. No. 1-4 oral solution based on the finished drawing.

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

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

Repeat steps 116, 127.

Lesson four.

Solving support problems for finding the elements of a regular triangular pyramid, problems for applying the properties of pyramids that have equal lateral edges and equal apothems.

Goals: to teach how to apply theoretical knowledge in solving elementary problems.

During the classes:

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

No. 11. table 5. (1.3)

1. Consider the properties of pyramids that have equal lateral edges and equal apothems. Group work. Collection of problems. No. 6, 7,8,9.
2. Homework: Collection of problems. No. 10 table 3. (2.4) table 4. (3.4)

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

Lesson five.

Problem solving.

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

During the classes:

1. Checking homework. Updating knowledge. Repetition of the properties of perpendicularity in space.
2. Problem solving. Collection of problems. No. 13, 14, 15.
3. Self-testing 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 problems. No. 12, 16.

Lesson six.

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

Goals: 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 have no questions complete assignments from E.M. Rabinovich Table 11.12.
2. Updating knowledge. 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 seminar topic, distribution of tasks into groups.
5. Homework: No. 71, prepare for the seminar.

Lesson seven.

Seminar on the topic: “Pyramid”

Goals: to help students develop the ability to use additional literature, the ability to put forward hypotheses and prove them, and develop 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.

In preparation for the seminar, each group works through the relevant sections of the textbook and lecture, and also uses additional literature and receives consultations from the teacher.

Seminar plan:

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

2. Regular polyhedra. Development of a regular polyhedron. (2nd group)

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

4. Homework: D/m A.P. Ershova. S-10, Option A 1 No. 1, S-11, Option B 1 No. 1, 2.

Lesson eight.

Lecture-presentation on the topic: “Regular polyhedra.”

Lecture outline

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

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. Use of the shape of regular polyhedra by nature and man.

6. Star polyhedra, their types.

7. Archimedean bodies, their types.

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

Lesson nine.

Test on the topic: “Pyramid”

Objectives: to test children's knowledge on this topic.

1 option №1. The side of the base of a regular quadrangular pyramid is 4 cm, and the apothem forms an angle of 60 with the plane of the base. 0 . find: a) the height of the pyramid; №2. The base of the pyramid is a regular triangle. Two side faces of the pyramid are perpendicular to the base, and the third is inclined at an angle of 30 0 . The height of the pyramid is 6cm №3. The base of the pyramid is a right triangle with legs 6 and 8 cm. All dihedral angles at the base of the pyramid are 60 0 . Find the total surface of the pyramid.

Option 2 №1. The height of a regular quadrangular pyramid is 4 cm, and the apothem forms an angle of 45 with the height 0 . find: a) the area of ​​the base of the pyramid; b) the lateral surface of the pyramid. №2. The base of the pyramid is a regular triangle with a side of 6 cm. Two side faces of the pyramid are perpendicular to the base, and the third is inclined at an angle of 60 0 . Find the total surface of the pyramid. №3. The base of the pyramid is a rectangle with sides 12 and 16 cm. The height of the pyramid is 24 cm and forms equal angles with all the side edges. Find the total surface of the pyramid.

Collection of problems 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 rib to the base plane.

3. Find 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 side rib to the plane of the base?

Table 1

					S side
				45°
				30°

table 2

					60°
					45°

6. The base of the pyramid KAVSD 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 lateral faces of the pyramid form angles of 60 with the plane of the base 0 . Find the pyramid's surface area and height.

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

					45°
					60°

Table 4

					45°

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

13 . In a regular quadrangular pyramid, the lateral edge is 20 cm and 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 ABCD. The height of the pyramid passes through vertex B. Prove that all the lateral faces of the pyramid are right triangles.

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

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

Literature

1. R.B. Reichmist. Problem book in mathematics for secondary school students and those entering universities. – M.: “Moscow Lyceum”, 2001.
2. L.O. Denishcheva, T.F. Mikheeva. Learning to solve problems. – M.: “Intelligence Center”, 2000.
3. T.L. Afanasyeva, L.A. Tapilina. “Geometry 11th grade” (lesson plans) - Volgograd: “Teacher”, 1999.
4. T.N. Mishchenko Workbook. Grade 11. To the textbook L.S. Atanasyan and others. Geometry grades 10-11.