Lesson presentation: "Stereometry". Basics of stereometry Presentation on the subject of stereometry axioms
Slide 1
Methodical development Savchenko E.M. MOU gymnasium №1, Polyarnye Zori, Murmansk region.
Subject of stereometry
Stereometry axioms
Geometry grade 10
Slide 2
Planimetry
Stereometry
Explores the properties of geometric shapes on a plane
Explores the properties of shapes in space
Translated from Greek, the word "geometry" means "surveying" "geo" - in Greek land, "metreo" - to measure
The word "stereometry" comes from the Greek words "stereos" volumetric, spatial, "metreo" - to measure
Slide 3
Planimetry
Stereometry
Along with these figures, we will consider geometric bodies and their surfaces. For example, polyhedra. Cube, parallelepiped, prism, pyramid. Rotation bodies. Ball, sphere, cylinder, cone.
Basic shapes: point, line
Basic shapes: point, line, plane
Other shapes: segment, ray, triangle, square, rhombus, parallelogram, trapezoid, rectangle, convex and non-convex n-gons, circle, circle, arc, etc.
Slide 4
To designate points, we use uppercase Latin letters
For the designation of straight lines, we use lowercase Latin letters.
Or we denote a straight line with two capital Latin letters.
Slide 5
The planes will be denoted by Greek letters.
In the figures, the planes are indicated in the form of parallelograms. A plane as a geometrical figure should be imagined as extending indefinitely in all directions.
Slide 6
Slide 7
When studying spatial figures, in particular geometric bodies, use their flat images in the drawing. The image of a spatial figure is its projection onto a particular plane. The same figure allows for different images.
Various cone images
Slide 8
Stereometry is widely used in construction, architecture, mechanical engineering, geodesy, and in many other fields of science and technology.
When designing this machine, it was important to get such a shape so that when moving, air resistance was minimal.
Slide 9
Opera House Sydney
Danish architect Jorn Utzon was inspired by the sight of the sails.
Slide 10
Eiffel Tower Paris, Champ de Mars
Engineer Gustave Eiffel found an unusual shape for his project. The Eiffel Tower is very robust: a strong wind deflects its top by only 10-12 cm.In the heat, from uneven heating by the sun's rays, it can deviate by 18 cm.
Slide 11
18,000 iron parts are fastened with 2,500,000 rivets
Slide 12
The original idea for the construction of the tower was found by architects L. Batalov and D. Burdin with the participation of designer N. Nikitin. Metal cables are stretched inside the cylindrical concrete blocks. This design is extremely stable.
The theoretical deflection of the tower top at maximum design wind speeds is about 12 meters.
Slide 13
The basic properties of points, lines and planes are expressed in axioms. We will formulate only three of the many axioms.
A1. Through any three points that do not lie on one straight line, a plane passes, and moreover, only one.
Illustration for Axiom A1: the glass plate will lie tightly on three points A, B and C, which do not lie on one straight line.
A
B
C
Slide 14
Illustrations to the A1 axiom from life.
The three-legged stool will always sit perfectly on the floor and will not sway. A four-legged stool has stability problems if the legs of the stool are not the same length. The stool sways, that is, it rests on three legs, and the fourth leg (the fourth "point") does not lie in the plane of the floor, but hangs in the air.
For a video camera, photography and other devices, a tripod is often used. The tripod's three legs will stably fit any indoor floor, on the asphalt or directly on the outdoor lawn, on the sand on the beach or in the grass in the forest. The three tripod legs will always find a plane.
Slide 15
O
A
V
Plotting right angles on the ground using a simple device called an ecker.
Tripod with ecker.
Slide 16
a
A2. If two points of a straight line lie in a plane, then all points of a straight line lie in this plane.
A
B
Slide 17
The property expressed in the A2 axiom is used to check the "evenness" of the drawing ruler. The ruler is applied with its edge to the flat surface of the table. If the edge of the ruler is even, then with all its points it adjoins the table surface. If the edge is uneven, then in some places a gap is formed between it and the table surface.
Slide 18
Axiom A2 implies that if a straight line does not lie in a given plane, then it has at most one common point with it. If a line and a plane have only one point in common, then they say that they intersect.
Slide 19
a
A3. If two planes have a common point, then they have a common straight line on which all the common points of these planes lie.
In this case, the planes are said to intersect in a straight line.
Slide 20
Axiom A3 is clearly illustrated by the intersection of two adjacent walls, a wall and a ceiling in a classroom.
Slide 21
A1. Through any three points that do not lie on one straight line, a plane passes, and moreover, only one.
Slide 22
Theorem
A plane passes through a straight line and a point not lying on it, and, moreover, only one.
M
a
Slide 23
Some consequences from the axioms.
Theorem
A plane passes through two intersecting lines, and moreover, only one
M
a
b
N
Slide 24
Training exercises
Name the planes in which the lines lie PE MK DB AB EC
P
E
A
B
C
D
M
K
Slide 25
Training exercises
Name the points of intersection of the straight line DK with the plane ABC, the straight line CE with the plane ADB.
P
E
A
B
C
D
The cycle of lessons on the topic: "Axioms of stereometry" consists of the following lessons:
1. Subject of stereometry. Axioms of stereometry "
2. Some derivations from the axioms.
3; 4. Solving problems on the application of axioms and their consequences.
5. Solving problems on the application of the axioms of stereometry and their consequences. Independent work.
A presentation has been prepared for each lesson.
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Cycle of lessons on the topic: "The axioms of stereometry and their consequences."
Lesson 1. Subject of stereometry. Stereometry axioms.
Lesson objectives:
- to familiarize students with the content of the stereometry course;
- study the axioms about the mutual arrangement of points, lines and planes in space;
- to learn to apply the axioms of stereometry when solving problems.
During the classes:
Slide 1.
1. Organizational moment.
2. Learning new material.
Teacher: For three years now, starting from the 7th grade, we have been studying the school geometry course.
Slide 2. Questions to students:
What is geometry? (Geometry is the science of the properties of geometric shapes)
What is planimetry? (Planimetry is a section of geometry that studies the properties of figures on a plane)
What basic concepts of planimetry do you know? (point, line)
Teacher: Today we are starting to study a new section of geometry - stereometry.
Slide 3. Stereometry is a section of geometry in which the properties of figures in space are studied. (Students write in a notebook)
Slide 4. Basic concepts of space: point, line, plane.
The idea of a plane is given by the smooth surface of a table, wall, floor, ceiling, etc. The plane, as a geometric figure, must be imagined stretching in all directions, infinite. The planes are designated by the Greek letters α, β, γ, etc.
1. Name the points lying in the plane β; not lying in the plane β.
2. Name the lines: lying in the plane β; not lying in the plane β.
Slide 5. We have a visual representation of the basic concepts (point, line, plane) and they are not given definitions. Their properties are expressed in axioms.
Along with a point, a straight line, a plane in stereometry, geometric bodies (cube, parallelepiped, cylinder, tetrahedron, cone, etc.) are considered, their properties are studied, their areas and volumes are calculated. The objects around us give an idea of geometric bodies.
Slide 6. Questions to students:
What geometric bodies do the objects depicted in these drawings remind you of?
Name objects in your environment (our classroom) that remind you of geometric bodies.
Slide 7. Practical work (in notebooks)
1. Draw a cube in a notebook (visible lines - a solid line, invisible - a dotted line).
2. Designate the vertices of the cube with capital letters ABCDA 1 B 1 S 1 D 1
3. Highlight with a colored pencil:
- vertices A, C, B 1, D 1 ; segments AB, SD, B 1 S, D 1 WITH; the diagonals of the square AA 1 IN 1 V.
Draw the attention of students to visible and invisible lines in the drawing; AA square image 1 in 1 In in space.
Slide 8. Questions for students:
What is an axiom? What planimetry axioms do you know?
In space, the basic properties of points, lines and planes, concerning their relative position, are expressed in axioms.
Slide 9. Students take notes and drawings in notebooks.
Axiom 1. (A1) Through any 3 points that do not lie on one straight line, there is a plane and, moreover, only one.
Slide 10. Note that if we take not 3, but 4 arbitrary points, then not a single plane may pass through them, that is, 4 points may not lie in the same plane.
Slide 11. Axiom 2. (A2) If 2 points of a straight line lie in a plane, then all points of a straight line also lie in this plane. In this case, they say that a straight line lies in a plane or a plane passes through a straight line.
Slide 12. Question to students:
How many points do a line and a plane have in common? (Fig. 1 - infinitely many; Fig. 2 - one)
Slide 13. Axiom 3. (A3) If two planes have a common point, then they have a common line on which all the common points of these planes lie.
In this case, the planes are said to intersect in a straight line.
3. Consolidation of the studied material.
Slide 14. Solving problems from textbook No. 1 (a, b), 2 (a).
Students read the statement of the problems and, using the picture on the slide, give an answer with an explanation.
Objective 1.
a) P, E (ADV) PE (ADV) according to A 2
Similar to MK (VDS)
V, D (ADV) and (VDS) VD (ADV) and (ICE)
Similar to AB (ADV) and (ABC)
C, E (ABC) and (DES) CE (ABC) and (DES)
b) C (DK) and (ABC) DK ∩ (ABC) = S. T. to. there are at most one point of intersection of a straight line and a plane (the straight line does not lie in the plane), then this is the only point.
Similarly, CE ∩ (ADV) = E.
Problem 2 (a)
In the plane DSS 1: D, S, S 1, D 1 , K, M, R. In the plane BQC: B 1, B, P, Q, C 1, M, C.
Slide 15. 4. Summing up the lesson.Questions to students:
- What is the name of the section of geometry that we will study in grades 10-11?
- What is Stereometry?
- Formulate with the help of the picture the axioms of stereometry that you learned in today's lesson.
Slide 16. 5. Homework.
Lesson 2. Some consequences from the axioms.
Lesson objectives:
Review the axioms of stereometry and their application in solving homework problems;
To acquaint students with the implications of the axioms;
To teach how to apply the corollaries from the axioms when solving problems, as well as to consolidate the ability to apply the axioms of stereometry when solving problems;
Repeat the formula for calculating the area of a rhombus.
During the classes.
Slide 1. 1. Organizational moment.Communication of the topic and objectives of the lesson.
Slide 2.
1) Formulate the axioms of stereometry and draw up the drawings on the board.
2) No. 1 (c, d); 2 (b, e).
Students orally from the picture on the slide answer the homework questions.
Slide 3. 3. Learning new material.Consider and prove the corollaries of the axioms.
Theorem 1. A plane passes through a straight line and a point not lying on it, and, moreover, only one.
Students write down the wording in a notebook and, answering the teacher's questions, make appropriate notes and drawings in the notebook.
What is given in the theorem? (straight and non-lying point)
What needs to be proven? (passes the plane; one)
What can be used to prove? (axioms of stereometry)
Which of the axioms allows you to build a plane? (A1, a plane passes through three points and, moreover, only one)
What is in this theorem and what is missing to use A1 (we have - a point; two more points are needed)
Where are we going to plot two more points? (on this line)
What conclusion can we draw? (build a plane through three points)
Does this plane belong to a straight line? (Yes)
On what basis can such a conclusion be drawn? (based on A2: if two points of a straight line belong to the plane, then the whole straight line belongs to the plane)
How many planes can you draw through a given line and a given point? (one)
Why? (since a plane passing through a straight line and a plane passes through a given point and two points on a straight line, it means that along A1 this plane is the only one)
Slide 4. Theorem 2. A plane passes through two intersecting lines and, moreover, only one.
Students prove the theorem on their own, then listen for several proofs and make additions and refinements (if necessary)
Pay attention to the fact that the proof is based not on axioms, but on Corollary 1.
Slide 5. 4. Consolidation of the studied material.
Problem 6 (from tutorial)
Students work in exercise books, suggest their own solutions, then compare their solution with the solution on the screen. Two cases are analyzed: 1) the points do not lie on one straight line; 2) the points are collinear.
Slide 6.7. The task is on the slide. Students read the condition, make a drawing and make the necessary notes in notebooks. The teacher conducts frontal work with the class on the problem. In the course of solving the problem, we repeat the formulas for calculating the area of a rhombus.
Given: AVSD - rhombus, AS∩VD = O, M, (A, D, O); AB = 4cm, A = 60º.
Find: (B, C); D (MOU); (MOB) ∩ (ADO); S AVSD.
Solution:
Pay attention to the fact that if two planes have common points, then they intersect in a straight line passing through these points.
5. Summing up:
Formulate the axioms of stereometry.
Formulate the consequences of the axioms.
The goal of the lesson has been achieved. We repeated the axioms of stereometry, got acquainted with the consequences of the axioms and applied them to solving problems.
Marking (with comments)
Slide 8. 6. Setting homework:
Lesson 3. Solving problems on the application of the axioms of stereometry and their consequences.
Lesson objectives:
Review the axioms of stereometry and their consequences;
To form the skill of applying the axioms of stereometry and their consequences when solving problems;
Students know the axioms of stereometry and their consequences and are able to apply them in solving problems.
During the classes.
Slide 1. 1. Organizational moment.Communication of the topic and objectives of the lesson.
2. Actualization of students' knowledge.
1) Checking homework on student questions.
Before the lesson, take homework notebooks from several students for review.
2) Two students are preparing at the blackboard the proof of the consequences of the axioms.
3) Two students (level 1) and two students (level 2) work on individual survey cards. Slide.
4) Frontal work with students.
Slide 2. Given: cube AVSDA1V1S1D1
Find:
- Several points that lie in the α plane; (A, B, C, D)
- Several points that do not lie in the α plane; (A 1, B 1, C 1, D 1)
- Several straight lines that lie in the plane α; (AB, VS, SD, AD, AS, VD)
- Several lines that do not lie in the plane α; (A 1 B 1, B 1 C 1, C 1 D 1, A 1 D 1, A 1 C 1, B 1 D 1, AA 1, BB 1, SS 1, DD 1)
- Several straight lines that intersect the line BC; (BB 1, CC 1)
- Several straight lines that do not cross the line BC. (HELL, AA 1 …)
Slide 3. Fill in the blanks to get the correct statement:
Slide 4. Are straight AAs 1 , AB, AD in the same plane? (Direct AA 1 , AB, HELL pass through point A, but do not lie in the same plane)
3. Solving problems.
Slide 5. Students solve problems No. 7, 10, 14 from the textbook, making appropriate drawings and notes on the board and in notebooks.
Problem number 7.
2) Do all the straight lines passing through the point M lie in the same plane?
Solution: By corollary 2:
2) All lines passing through the point M do not necessarily lie in the same plane. (see example from slide 4)
Problem 10. Students solve the problem on their own (similar to problem number 7). The teacher selectively takes notebooks for checking and provides individual assistance in solving the problem to students who did not cope with the task.
Problem number 14. Solution: All lines a, b, c lie in the same plane. In this case, by Corollary 2, a plane can be drawn, and one plane passes through three straight lines.
One of the three straight lines, for example, c, does not lie in the plane α defined by straight lines a and b. In this case, three different planes pass through the given three straight lines, defined by pairs of straight lines a and b, a and c, b and c.
Slide 6. Students make a drawing and the necessary constructions and notes in notebooks. When building, students pronounce the axioms, the result of the construction is written using symbols.
Task. Given: cube AVSDA 1 B 1 C 1 D 1
tM lies on the edge BB 1 , point N lies on the edge CC 1 and point K lies on the edge DD 1
a) Name the planes in which the points M lie; N.
b) find the point F-point of intersection of lines МN and BC. What property does point F have?
c) find the point of intersection of the straight line KN and the plane ABC.
d) find the line of intersection of the planes MNK and ABC.
Solution:
Slide 7. To solve the next problem, we repeat the formula for calculating the area of a quadrilateral. The derivation of the formula is parsed across the slide.
Students write the formula down in a notebook.
Slide 8. Prove that all the vertices of the AVSD quadrilateral lie in the same plane if its diagonals AC and VD intersect.
Calculate the area of the quadrangle, if AS┴VD, AS = 10cm, VD = 12cm.
Answer: 60 cm 2
4. Summing up the lesson.
What caused the difficulty? The teacher announces the grades for the lesson with a commentary.
Slide 9.
Lesson 4. Solving problems on the application of the axioms of stereometry and their consequences.
Lesson objectives:
Conduct control of knowledge of the axioms of stereometry and their consequences;
To consolidate the formed skill of applying the axioms of stereometry and their consequences when solving problems;
Review: Pythagorean theorem and its application; formulas for calculating the areas of an equilateral triangle, rectangle.
During the classes.
Slide 1. 1. Organizational moment.Communication of the topic and objectives of the lesson.
Slide 2. 2. Checking homework.
Before the lesson, take homework notebooks from several students for review.
Two students are preparing at the blackboard solutions to problems from homework - No. 9, 15.
The rest of the students answer the questions of the math dictation on the slide.
Slide 3. 3. Problem solving (frontal work with the class)
Problem number 1.
You are given a MABS tetrahedron, each edge of which is 6 cm.
- Name the line along which the planes intersect: a) MAB and MFC; b) MCF and ABC.
- Find the length of СF and SАВС
- How to build the point of intersection of the straight line DE with the plane ABC?
Questions for students (if necessary):
Which points belong to both planes at the same time. What axiom can be used to draw a conclusion?
State the property of the median of an isosceles triangle.
Formulate the Pythagorean theorem.
Why can we apply the Pythagorean theorem in this case?
What methods can be used to calculate the area of an equilateral triangle?
Is it always possible to build the point of intersection of the straight line MU with the plane ABC?
Slide 4. Problem number 2.
- How to build the point of intersection of the plane ABC with the straight line D 1 R?
- How to draw a line of intersection of the plane of blood pressure 1 P and ABB 1?
- Calculate the length of the segments AR and BP 1 if AB = a
Solution:
Slide 5. Problem number 3.
Given : Points A, B, C do not lie on one straight line.
Prove that point P lies in the plane ABC.
With the help of animation on the slide, students draw the appropriate constructions and the necessary conclusions. They make notes in notebooks using mathematical symbols, pronouncing the corresponding axioms and consequences from the axioms.
Questions to students (as needed):
Knowing that points A, B, C do not lie on one straight line, what conclusion can be drawn?
If points A and B lie in the plane, what conclusion can be drawn about line AB?
What conclusion can be drawn about the point M?
If points A and C lie in the plane, what conclusion can be drawn about the line AC?
What conclusion can be drawn about the point K?
Knowing that the points M and K lie in the plane, what conclusion can be drawn about the straight line MK?
What conclusion can be drawn about the point P?
Solution (another way of proving):
AB∩AC = A. According to the second corollary, straight lines AB and AC define the plane α. Point M belongs to AB, which means it belongs to the plane α, and point K belongs to AC, and therefore also to the plane α. According to axiom A2: MK lies in the plane α. The point P belongs to the MC, and hence to the plane α.
Slide 6. Problem number 4.
The planes α and β intersect in a straight line with. Line a lies in the plane α and intersects the plane β. Do lines a and c intersect? Why?
Questions to students (if necessary):
Knowing that the straight line a intersects the plane β, what conclusion can be drawn? (A straight line and a plane have a common point, for example, point B)
What property does point B have? (Point B belongs to line a, plane α, and plane β)
If a point belongs to two planes at the same time, then what can we say about the relative position of the planes? (planes intersect in a straight line, for example with)
What is the relative position of point B and line c? (point B belongs to line c)
Knowing that point B belongs to both line a and line c, what conclusion can be drawn about these lines? (lines intersect at point B)
Slide 7. Problem number 5.
Given a rectangle AVSD, O is the point of intersection of its diagonals. It is known that points A, B, O lie in the plane α. Prove that points C and D also lie in the plane α. Calculate the area of the rectangle if AC = 8 cm, AOB = 60º.
The task is intended for an independent solution with a discussion of the solution and the provision of individual assistance to students. It is helpful to discuss different ways of finding the area of a rectangle:
Invite students to solve the problem in different ways. Answer: 16 cm 2.
4. Summing up the lesson:
What axioms and theorems did we use in the lesson when solving problems? Formulate.
What tasks were the most interesting, the most difficult?
What was useful for you personally in the lesson?
What caused the difficulty?
Marking for the lesson (with commenting on each mark)
Slide 8. 5. Setting homework:
Lesson 5. Solving problems on the application of the axioms of stereometry and their consequences. Independent work (20 min.)
Lesson objectives:
To consolidate the assimilation of theoretical questions in the process of solving problems;
Check the level of preparedness of students by conducting independent work of a controlling nature.
During the classes.
Slide 1. 1. Organizational moment.
Communication of the topic and objectives of the lesson.
Slide 2. 2. Checking homework.
Before the lesson, take homework notebooks from several students for review.
Objective 1.
Lines a and b intersect at point O, A a, B b, P AB. Prove that lines a and b and point P lie in the same plane.
Solution:
Slide 3. Task 2.
In this figure, the plane α contains points A, B, C, D, but does not contain point M. Construct point K - the point of intersection of line AB and plane MSD. Does point K lie in the plane α.
Solution:
Slides 4, 5, 6 3. Verbal solution of problems for theory revision (by slides)
Slides 7.8 4. Independent work(multilevel, controlling nature) Students choose their level of difficulty.
5. Summing up.
1) Collect notebooks with independent work.
2) Announcement of marks with comments.
Slide 9. 6. Homework.
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Slide captions:
Lesson 1 Topic: "Subject of stereometry. Axioms of stereometry."
What is geometry? Geometry - the science of the properties of geometric shapes "Geometry" - (Greek) - "surveying" - What is planimetry? Planimetry is a section of geometry in which the properties of figures on a plane are studied. And a Basic concepts of planimetry: point is a straight line - Basic concepts of planimetry?
Stereometry - a section of geometry in which the properties of figures in space are studied
Basic figures in space: point straight plane α β Designation: А; V; WITH; ...; М;… a А В М N Р Designation: a, b, c, d ..., m, n, ... (or in two capital letters in Latin) Designation: α, β, γ ... Answer the questions in the figure: 1. Name the points, lying in the plane β; not lying in the plane β. 2. What are the lines lying in the plane β; not lying in the plane β
Some geometric bodies. A B C D D 1 C 1 B 1 A 1 cube A B C D A 1 B 1 C 1 D 1 parallelepiped A B C D tetrahedron cylinder cone
Name which geometric bodies remind you of the objects depicted in these figures: Name the objects from your environment (our classroom) that remind you of geometric bodies.
Practical work. 1. Draw a cube in a notebook (visible lines - a solid line, invisible - a dotted line). 2. Designate the vertices of the cube with capital letters ABCDA 1 B 1 C 1 D 1 A B C D D 1 C 1 B 1 A 1 3. Select with a colored pencil: vertices A, C, B 1, D 1 segments AB, SD, B 1 С, Д 1 С square diagonal АА 1 В 1 В
What is an axiom? An axiom is a statement about the properties of geometric figures, it is taken as the starting point, on the basis of which theorems are proved further and, in general, all geometry is constructed. Planimetry axioms: - through any two points you can draw a straight line and, moreover, only one. of three points, one straight line, and only one, lies between the other two. there are at least three points that do not lie on one straight line ...
Stereometry axioms. A B C A1. Through any three points that do not lie on one straight line, a plane passes and, moreover, only one. α
If the legs of the table are not the same in length, then the table stands on three legs, i.e. rests on three "points", and the end of the fourth leg (fourth point) does not lie in the plane of the floor, but hangs in the air.
Stereometry axioms. А В α А2. If two points of a straight line lie in a plane, then all points of this straight line lie in this plane. They say: a straight line lies in a plane or a plane passes through a straight line.
a M The straight line lies in the plane. The straight line intersects the plane. How many points do the line and the plane have in common?
Stereometry axioms. α β A3. If two planes have a common point, then they have a common straight line on which all the common points of these planes lie. They say: planes intersect in a straight line. Ah
Solve problems: №1 (a, b); 2 (a) A B S D R E K M A B S D A 1 B 1 C 1 D 1 Q P R K M Name according to the figure: a) the planes in which the straight lines DV, AB, MK, PE, EC lie; b) the point of intersection of the straight line DC with the plane ABC, the straight line CE with the plane ADV. a) points lying in the planes DSS 1 and B Q C No. 1 (a, b) No. 2 (a)
Let's summarize the lesson: 1) What is the name of the geometry section that we will study in grades 10-11? 2) What is stereometry? 3) Formulate with the help of the picture the axioms of stereometry that you learned in the lesson today. А А В В α α А α β
Theorem 1. A plane passes through a straight line and a point not lying on it, and, moreover, only one. Given: a, M ¢ a Prove: (a, M) with α α is the only a M α Proof: 1. P, O with a; (P, O, M) ¢ a P O According to the A1 axiom: a plane passes through the points P, O, M. According to Axiom A2: since two points of a straight line belong to a plane, then the whole straight line belongs to this plane, i.e. (a, M) with α 2. Any plane passing through a straight line a and a point M passes through points P, O, and M, which means, according to the A1 axiom, it is the only one. Ch.t.d. Some consequences from the axioms:
Theorem 2. A plane passes through two intersecting lines, and, moreover, only one. Given: a ∩ b Prove: 1. (a∩ b) c α 2. α is the only a b M H α Proof: 1. The plane α passes through a and H a, H b. (M, H) α, (M, H) b, so by A2 all points b belong to the plane. 2. The plane passes through a and b and it is the only one, since any plane passing through lines a and b also passes through H, which means that α is the only one.
Solve problem number 6 А В С α Three given points are connected in pairs by segments. Prove that all line segments lie in the same plane. Proof: 1. (A, B, C) α, so along A1 through A, B, C the only plane passes. 2. Two points of each segment lie in the plane, hence, according to A2, all points of each of the segments lie in the plane α. 3. Conclusion: AB, BC, AC lie in the plane α 1 case. A B C α 2 case. Proof: Since 3 points belong to one straight line, then according to A2, all points of this straight line lie in the plane.
Task. А В С Д М О AVSD - rhombus, O - point of intersection of its diagonals, M - point of space that does not lie in the plane of the rhombus. Points A, D, O lie in the plane α. Determine and substantiate: Do points B and C lie in the plane α? Does point D lie in the MOV plane? Name the line of intersection of the MOV and ADO planes. Calculate the area of a rhombus if its side is 4 cm and the angle is 60 º. Suggest different ways to calculate the area of a rhombus.
Oral work. A B S D A 1 B 1 C 1 D 1 α Given: cube ABSDA 1 B 1 C 1 D 1 Find: Several points that lie in the plane α; Several points that do not lie in the α plane; Several straight lines that lie in the plane α; Several lines that do not lie in the plane α; Several straight lines that intersect the line BC; Several straight lines that do not cross the line BC. Objective 1.
Oral work. Problem 2. α А М В а b c Fill in the blanks to get the correct statement:
Oral work. A B C D A 1 B 1 C 1 D 1 α Lines AA 1, AB, HELL pass through point A, but do not lie in the same plane Do lines AA 1, AB, HELL lie in the same plane?
Solve the problems from the tutorial: pp. 8 № 7, 10, 14. Students' work on the board and in notebooks:
Problem 1 A B C D A 1 B 1 C 1 D 1 M NF K Given: the cube ABCDA 1 B 1 C 1 D 1 tM lies on the edge BB 1, i.e. N lies on the edge CC 1 and point K lies on edge DD 1 a) name the planes in which the points M lie; N. b) find the point F-point of intersection of lines M N and BC. What property does point F have? c) find the point of intersection of the straight line K N and the plane ABC O d) find the line of intersection of the planes M N K and ABC
Problem (orally) A B C D M O AVSD is a rhombus, O is the point of intersection of its diagonals, M is a point in space that does not lie in the plane of the rhombus. Points A, D, O lie in the plane α. Determine and substantiate: 1. What other points lie in the plane α? Do points B and M lie in the plane α? Does point B lie in the plane of the MOD? Name the line of intersection of the planes MOC and ADO. Point O is the common point of the MOV and MOC planes. Is it true that these planes intersect in a straight line MO? Name three straight lines lying in the same plane; not lying in the same plane.
Problem (orally) A B CM Sides AB and AC of triangle ABC lie in the plane. Prove that the median also lies in the plane.
S D V E F O M Task (oral) What is the error of the drawing, where O E F. Give an explanation. What a correct drawing should look like.
1 level A B C S K M N 1. Using this figure, name: a) four points lying in the plane S AB; b) the plane in which the straight line M N lies; c) a straight line along which the planes S AC and S BC intersect. 2. Point C is the common point of the plane and. Line c passes through point C. Is it true that the planes and intersect along the line c. Explain the answer. 3. Through line a and point A, two different planes can be drawn. What is the relative position of line a and point A. Explain the answer. 2 level S А В С Д Е F 1. Using this figure, name: a) two planes containing the straight line DE; b) the straight line along which the planes AE F and S BC intersect; c) planes intersected by line S B. 2. Lines a, b and c have a common point. Is it true that these lines lie in the same plane? Justify the answer. 3. Planes and intersect in a straight line with. Line a lies in the plane and intersects the plane. What is the relative position of straight lines a and c?
A B C D A 1 B 1 C 1 D 1 Level 3 (on cards) 1. Using this figure, name: a) two planes containing line B 1 C; b) a straight line along which planes В 1 СД and АА 1 Д 1 intersect; c) a plane that does not intersect with the straight line SD 1. 2. Four straight lines intersect in pairs. Is it true that if any three of them lie in the same plane, then all four lines lie in the same plane? Explain the answer. 3. Vertex C of the plane quadrilateral AVSD lies in the plane, and points A, B, D do not lie in this plane. Straight lines AB and HELL intersect the plane at points B 1 and D 1, respectively. What is the relative position of points C, B 1 and D 1? Explain the answer.
Homework: repeat the material from planimetry and make notes in notebooks on the following questions: Determination of parallel lines Mutual position of two lines on a plane Construction of a line parallel to a given Axiom of parallel lines
1st lesson: What does stereometry study? Stereometry is a branch of geometry that studies the properties of figures in space. The word "stereometry" comes from the Greek words "stereos" - volumetric, spatial and "metreo" - to measure. Many geometric terms have been translated from the ancient Greek language, because geometry originated in ancient Greece and developed in schools of philosophy.
2nd lesson: Basic shapes of stereometry. There are various ways of depicting a plane: the plane is depicted with a parallelogram; a plane is indicated by a figure bounded by two parallel straight lines and two arbitrary curves; the plane is transmitted by a figure of any shape.
3rd lesson: Spatial figures. The lesson is devoted to preparation for the introduction of the axioms of stereometry. Students are offered the following tasks: 1. Draw a line a, a point A lying on it and a point B not lying on it. 2. Draw a plane and two intersecting lines a and b lying on it. 3. Draw the plane, the points A and B lying on it, as well as the points C and D located on opposite sides of the plane. 4. Draw the plane and the line intersecting it a. 5. Draw planes intersecting at right angles.
5th lesson: Signs of plane parallelism. When studying the axioms of stereometry, we recall the first axioms of planimetry and formulate their spatial analogies. As a result, we get the following table: Ax uom a Drawing Formulation P1P1 Whatever the line in space, there are points in space that belong to this straight line, and points that do not belong to it. P2P2 Through any two points in space, you can draw a straight line, and moreover, only one.
6th lesson: Parallel design. Consider the consequences of the axioms: Drawing Formulation Sl.1 Through a straight line and a point not lying on it, you can draw a plane, and moreover, only one. If two points of a straight line belong to a plane, then the whole straight line belongs to this plane. Through three points that do not lie on one straight line, you can draw a plane, and, moreover, only one.
Depiction of spatial figures on a plane There are seven lessons on the topic: 1. Parallel design and its main properties; 2. P Parallel design of plane figures; 3. And the Image of spatial figures in parallel projection; 4. C Section of polyhedra; 5. З Golden section; 6. Central design and its properties; 7. And The image of spatial figures in the central projection.
Lesson 1: Parallel design and its basic properties. The main properties of parallel design: 1. A parallel projection of a straight line is a straight line or a point; 2. a parallel projection of a line segment is a line segment or a point; 3. the ratio of the lengths of the segments lying on one straight line is preserved (in particular, the midpoint of the segment with parallel projection goes to the midpoint of the corresponding segment); 4. Parallel projection of two parallel lines are parallel lines, or one line, or two points; 5. the ratio of the lengths of the segments lying on parallel lines is preserved during parallel design; 6. If the figure lies in a plane parallel to the projection plane, then its parallel projection onto this plane will be a figure equal to the original one.
Lesson 2: Parallel projections of plane figures. The question of the image of flat figures in parallel design is considered. Students should imagine which shapes are parallel projections of polygons and a circle. Find out what properties of polygons are preserved during parallel design. Learn how parallel projections of basic planar figures are constructed.
The golden ratio in architecture The famous Russian architects M. Kazakov and V. Bazhenov widely used the golden ratio in their work. For example, the golden ratio can be found in the architecture of the Senate building in the Kremlin. According to the project of M. Kazakov, the First Clinical was built in Moscow. Another architectural masterpiece of Moscow - the Pashkov House - is one of the most perfect pieces of architecture by V. Bazhenov.
Polyhedra. This course includes the following activities: 1. Regular polyhedra. Regular polyhedra. 2. Semiregular polyhedra. Semiregular polyhedra. 3. Star polyhedra. Star polyhedra. 4. Euler's theorem. Euler's theorem.
Lesson 4: Euler's theorem. One of the most interesting properties of convex polyhedra is described by Euler's theorem. Name of polyhedron a Number of vertices (B) Number of edge p (P) Number of faces (D) Triangular pyramid 464 Quadrangular prism 8126 Pentagonal bipyramid regular dodecahedron n-angle pyramid n + 12n2n n-angle prism 2n2n3n3nn + 2 First with students consider the polyhedra known to them and fill in the table. Then the theorem itself is deduced: В-Р + Г = 2
Angles between lines and planes in space. When studying this topic, it is desirable to note that the problem of measuring angles dates back to ancient times. The history of the creation of measuring instruments and methods of measurement should be covered as widely as possible. For this, it is proposed to conduct the following classes: 1. The volume of figures in space. Cylinder volume; The volume of figures in space. Cylinder volume; 2. Cavalieri principle; Cavalieri principle; 3. The volume of the cone; Cone volume; 4. Volume of the sphere. The volume of the ball.
Lesson 1: The volume of figures in space. Cylinder volume. This lesson discusses the problems of measuring the volumes of spatial figures. The main properties of the volume are enumerated: ooooooooooooobrazovaniya figure in space is a non-negative number; oooo the volume of a cube with edge 1 is 1; oraral figures have equal volumes; oeo if figure Ф is composed of figures Ф 1 and Ф 2, then the volume of figure Ф is equal to the sum of volumes of figures Ф 1 and Ф 2.
- What is Stereometry?
- The emergence and development of stereometry
- Basic figures in space
- Designation of points and examples of their models
- Line designation
- Examples of models of straight lines
- Designation of planes and examples of their models
- What else does stereometry study?
- Objects and geometric bodies around us
- Displaying geometric bodies in drawings
- Practical (applied) value of stereometry
- Stereometry axioms
- Consequences from the axioms of stereometry
- Anchoring
- Used Books
What is Stereometry?
Stereometry Is a section of geometry in which the properties of figures in space are studied.
The emergence and development of stereometry.
- The development of stereometry began much later than planimetry.
- Stereometry developed from observations and solutions to questions that arose in the process of human practical activity.
- Already primitive man, having taken up agriculture, made attempts to estimate, at least in rough terms, the size of the harvest he had harvested by the masses of grain piled up in heaps, heaps or stacks.
- The builder of even the most ancient primitive buildings had to somehow take into account the material that he had at his disposal, and also be able to calculate how much material would be required to erect a particular building.
- Masonry among the ancient Egyptians and Chaldeans required familiarity with the metric properties of at least the simplest geometric bodies.
- The need for agriculture, navigation, orientation in time pushed people to astronomical observations, and the latter to study the properties of the sphere and its parts, and, consequently, the laws of the mutual arrangement of planes and lines in space.
Basic figures in space.
Plane - a geometric figure that extends indefinitely in all directions
Designation of points and examples of their models.
Points are designated by capital Latin letters A, B, C, ...
Examples of point models are:
atoms and molecules
planets in the scale of the universe
Designation of straight lines.
- Straight lines are designated:
- lowercase latin letters a, b, c, d, e, k, ...
- two capital Latin letters AB, CD ...
Examples of models of lines.
Examples of line models are:
airplane contrails
Designation of planes and examples of their models.
The planes are designated by the Greek letters α, β, γ, ...
Examples of plane models are:
water surface
table surface
What else does stereometry study?
Along with a point, a line and a plane, stereometry studies geometric bodies and their surfaces.
Objects and geometric bodies around us.
The objects around us give ideas about geometric bodies.
And by studying the properties of geometric figures - imaginary objects, we get information about the geometric properties of real objects and can use these properties in practice.
crystals-polyhedrons
tin can - cylinder
candy packaging - cone
Images of geometric bodies in drawings.
- The image of a spatial figure is its projection onto a particular plane.
- Invisible parts of the figure are drawn with dashed lines.
Practical (applied) value of stereometry.
- Geometric bodies are fictional objects
- Studying the properties of geometric shapes, we get ideas about the geometric properties of real objects (their shape, relative position, etc.)
- Stereometry is widely used in construction, architecture, mechanical engineering and other fields of science and technology.
Stereometry axioms.
- Axiom- this statement about the properties of geometric figures is taken as the starting point, on the basis of which theorems are proved further and in general all geometry is constructed.
Stereometry axioms.
A1 . Through any three points that do not lie on one straight line, a plane passes and, moreover, only one.
Stereometry axioms.
A2 ... If two points of a straight line lie in a plane, then all points of this straight line lie in this plane.
In this case, they say that a straight line lies in a plane or a plane passes through a straight line.
Stereometry axioms.
A3. If two planes have a common point, then they have a common straight line on which all the common points of these planes lie.
They say that planes intersect in a straight line
Consequences from the axioms.
Theorem 1: A plane passes through a straight line and a point not lying on it, and moreover, only one.
Theorem 2: A plane passes through two intersecting lines, and moreover, only one.
Anchoring.
1. Name the planes in which the lines lie:
Anchoring.
2. Name the point of intersection of the line CE with the plane ADB.
3. Name the lines along which the planes intersect:
Used Books
- Geometry. 10-11 grades: textbook. For general education. institutions: basic and profile. levels / HP Atanasyan, V.F. Butuzov, S.B. Kadomtsev et al. - 21st ed. - M .: Education, 2012.- 255 p .: ill.
- Geometry: Methodological Guide for Higher Pedagogical Institutions and Secondary School Teachers: Part 2 Stereometry / ed. Prof. I.K. Andronov.
The school geometry course consists of two parts:
PLANIMETRYSTEREOMETRY
Planimetry is a section
geometry in which
properties are studied
geometric shapes
on surface.
Stereometry is a section
geometry in which
properties are studied
geometric shapes
in space.
The word "stereometry" comes from the Greek
the words "stereos" - three-dimensional, spatial and
"Metreo" - to measure.
2
Basic concepts
planimetryPoint
Straight
stereometry
Point
Straight
Plane
is a geometric figure,
extending indefinitely into all
sides.
3
Along with points, straight lines, planes in stereometry, geometric bodies are considered, their properties are studied, their areas are calculated.
Along with points, lines, planesin stereometry
geometric bodies are considered,
their properties are studied,
the areas of their surfaces are calculated,
and also the volumes of bodies are calculated.
cube
ball
cylinder
4
Volumetric geometric bodies
PolyhedraRotation bodies
prism
pyramid
cone
parallelepiped
cylinder
cube
ball
5
Points are designated by capital Latin letters A, B, C, D, E, K, ...
AV
WITH
E
Straight lines are indicated by lowercase
latin letters a, b, c, d, e, k, ...
b
d
a
Planes are designated by Greek
letters α, β, γ, λ, π, ω, ...
β
γ
α
6
Stereometry is widely used in the construction industry
7Stereometry is used in architecture
8Stereometry is used in mechanical engineering
9Stereometry is used in surveying
Geodesy is the science that studies the species andthe size of the earth.
In many other fields of science and technology.
10
It is clear that in each plane some points of space lie, but not all points of space lie in the same plane.
Aє, Bє,M
Mє, Nє, Pє
A
N
B
P
11
Stereometry axioms
Axiom 1Any three
points not
lying on one
straight, passes
plane, and
moreover, only
one.
A
V
WITH
Axiom 3
Axiom 2
If two
planes have
common point, then
they have
straight on
which all lie
common points of these
planes.
If two points
straight lie in
plane, then all
points of a straight line
lie in this
plane.
A
V
WITH
A
a
α
12
Some consequences of the axioms
Qα
a
P
M
Theorem 2. After two
intersecting straight lines
passes the plane, and
moreover, only one.
Theorem 1. Across the line
and not lying on it
the point passes the plane,
and, moreover, only one.
b
a
α
M
Spelling the suffixes of different parts of speech
Higher professional education
Rebus in Russian