Presentation for the lesson: "Stereometry". Presentation - the subject of stereometry - axioms of stereometry Download presentation on the topic of stereometry

- What is geometry?

Geometry is a branch of mathematics that studies spatial structures and relationships, as well as their generalizations.

“Geometry” - (from Greek) – “land surveying”.

• What is planimetry?

Planimetry is a section of geometry in which the properties of figures on a plane are studied.

- Basic concepts of planimetry?

Basic figures in space:

point straight plane

Designation: A; IN; WITH; ...; M;...

Designation: a, b, с, d…, m, n,…(or two capital Latin ones)

Designation: α, β, γ…

Name what geometric bodies the objects depicted in these pictures remind you of:

Name objects from your environment (our classroom) that remind you of geometric bodies.

1. Depict in the notebook there is a cube (visible lines are solid, invisible lines are dotted).

2. Designate vertices of the cube in capital letters ABCDA 1 B 1 C 1 D 1

3. Highlight colored pencil:

• vertices A, C, B 1, D 1
• segments AB, CD, B 1 C, D 1 C
• square diagonal AA 1 B 1 B

- What is an axiom?

An axiom is a statement about the properties of geometric figures; it is accepted as the starting point, on the basis of which further theorems are proved and, in general, all geometry is built.

Axioms of planimetry:

- through any two points you can draw a straight line and, moreover, only one.

• Of the three points on a straight line, one, and only one, lies between the other two.
• there are at least three points that do not lie on the same line...

Axioms of stereometry.

A1 . Through any three points that do not lie on the same line, there passes a plane, and only one.

Axioms of stereometry.

A2. If two points of a line lie in a plane, then all points of this line lie in this plane.

They say: the straight line lies in the plane or the plane passes through the line.

How many points do a line and a plane have in common?

The straight line lies in the plane

A straight line intersects a plane

Axioms of stereometry.

A3. If two planes have a common point, then they have a common line on which all the common points of these planes lie. They say : planes intersect in a straight line.

Solve problems: No. 1 (a, b); 2(a)

Name according to the picture:

IN 1

WITH 1

A 1

D 1

a) the planes in which the straight lines PE, MK, DV, AB, EC lie; b) the points of intersection of straight line DK with plane ABC, straight line CE with plane ADV.

a) points lying in the DSS planes 1 and BQC

Let's summarize the lesson:

1) What is the name of the section of geometry that we will study in grades 10-11?

2) What is stereometry?

3) Using a drawing, formulate the axioms of stereometry that you studied today in class.

• Review the axioms of planimetry
• Learn axioms A1-A3
• Read paragraph 1.2 (pages 3 – 6)
• Solve problems: 1 (c, d); 2(b,d).
• Additionally: No. 3; 4 (optional)

Stereometry

Stereometry. Pencil. Geometry. Planimetry. Basic concepts of stereometry. Axioms of stereometry. Axioms. Line points. Planes. Corollaries from the axioms. Intersecting lines. Plane. Determination of body volume. Bodies with equal volumes. Volume of a rectangular parallelepiped. Prism volumes. Two right triangles. Volume of an inclined prism. Perpendicular section. Polyhedron. Rectangles. Image planes. Parallelepiped. Rectangular parallelepiped. Pyramid. Tetrahedron. Figure. Segments. Truncated pyramid. Octahedron. Dodecahedron. Icosahedron. Cylinders. Bodies of rotation. Ball sector. - Stereometry.ppt

Basics of stereometry

On teaching stereometry in humanities classes. What does stereometry study? The angle between straight lines in space. Parallelepiped. Fourth quarter. Stereometry. Pythagoras. Basic figures of stereometry. Spatial figures. Parallelism of straight lines and planes. Signs of parallel planes. Parallel design. Image of spatial figures on a plane. Parallel design and its basic properties. Parallel projections of plane figures. Image of spatial figures. Section of polyhedra. Golden ratio. Golden ratio in sculpture. Golden ratio in architecture. - Basics of stereometry.ppt

Subject of stereometry

Axioms of stereometry. Geometry. Stereometry science concept. Visual representations. From the history. Stereometry. Egyptian pyramids. Do you remember the Pythagorean theorem? Pythagoras. Pythagorean theorem. Pentagram. Regular polyhedra. Universe. Philosophical school. Euclid. Spatial representations. Undefinable concepts. Basic concepts of stereometry. Invisible side. Planimetry. Dots. Directions. Today in class. - Subject of stereometry.ppt

Introduction to Stereometry

School geometry. Arithmetic. Geometric knowledge was applied. Geometric knowledge helped. Let's translate it into the language of squares. Let's take 6 matches. Plane. Planimetry. Crossword. Stereometry -. Polyhedron. Figures. Bodies. The mobile dwellings of the Indians are called Tipis. Magazine "Kvant". Summing up the lesson. - Introduction to stereometry.ppt

Axioms of geometry

Axioms of stereometry. Get acquainted with the axioms of stereometry. Planimetry. Dots. You can draw a straight line and only one. Of the three points, only one lies between the other two. Each segment has a certain length. A straight line divides a plane into two half-planes. Each angle has a certain degree measure. You can set aside a segment of a given length and only one. You can plot an angle on any half-line from the starting point. Triangle. You can draw at most one straight line on a plane. Stereometry. Axioms. Points in space. Different planes have a common point. You can draw a plane, and only one. - Axioms of Geometry.pptx

Axioms of stereometry

Axioms of stereometry. 1. Concepts of stereometry 2. Image of a plane 3. Axioms of stereometry 4. Corollaries from the axioms of stereometry. The system of axioms of stereometry consists of axioms of planimetry and three axioms of stereometry. Stereometry is a branch of geometry in which the properties of figures in space are studied. The picture shows two generally accepted images of a plane. Planes are designated by small Greek letters: a, b, g, ... There is at least one straight line and at least one plane. The distance from point A to point B is equal to the distance from point B to point A: AB=BA. Corollaries from the axioms of stereometry. - Axioms of stereometry.ppt

Axioms of stereometry grade 10

Axioms of stereometry. A, B, C? one straight line A, B, C? ? ? - the only plane. In any plane of space, all axioms and theorems of planimetry are valid. Corollaries from the axioms of stereometry. A plane passes through two intersecting lines, and only one. 1. Do they lie on a plane? points B and C? 2. Does point D lie on the plane (MOV)? 3. Name the line of intersection of the planes (MOV) and (ADO). Name the different ways to calculate the area of ​​a rhombus. The problem is the intersection of two planes ABCDA1B1C1D1 is a cube, K belongs to DD1, DK=KD1. Give answers to the questions below with the necessary justification. - Axioms of stereometry grade 10.ppt

Basic axioms of stereometry

Corollaries from the axioms of stereometry

Slides on geometry. Axioms of stereometry and some consequences from them. Stereometry. Planimetry. Geometry section. Axioms of stereometry. Different planes. Various straight lines. Axioms of planimetry. Construct an image of a cube. Explain your answer. The existence of a plane. Explanation of new material. Oral work. Find the line of intersection of the planes. Which planes does the point belong to? Plane. Proof. Elements of a cube. The intersection of a line and a plane. Flat and straight. How many faces pass through one, two, three, four points. Straight lines intersecting at a point. - Corollaries from the axioms of stereometry.ppt

Spatial figures on a plane

Image of spatial figures on a plane. The purpose of the lesson. True False. One of two parallel lines intersects a plane. By the plane intersection lemma. Is it true that two disjoint lines in space are parallel? Parallel and intersecting lines do not have common points. If two lines are parallel to a certain plane, then they are parallel to each other. Lines can not only be parallel, but also intersect. Two planes are intersected by two parallel lines. There are no conditions for fulfilling the plane parallelism test. Gerard Desargues. - Spatial figures on a plane.ppt

The relative position of lines in space

The relative position of lines in space. Crossing straight lines. Introduce the definition of skew lines. Introduce formulations and prove the sign and property of skew lines. Location of straight lines in space: They lie in the same plane! Given a cube ABCDA1B1C1D1. Are lines AA1 and DD1 parallel? AA1 and CC1? 2. Are AA1 and DC parallel? Sign of crossing lines. Given: AB?, CD? ? = C, C AB. Consolidation of the studied theorem: Determine the relative position of the lines AB1 and DC. 2. Indicate the relative position of the straight line DC and the plane AA1B1B. - The relative position of lines in space.ppt

Problems in stereometry

Tasks. Find the volume of the pyramid. Find the volume V of the cylinder. Find the surface area of ​​the polyhedron. Circumference. Find the area of ​​the trapezoid. Find the ordinate of point A. Find the angle of the polyhedron. Find the square of the distance between the vertices. Volume of the ball and its parts. Circular sector. Diameter of lead ball. - Problems on stereometry.pptx

“Geometry problems” 11th grade

Use of ICT. Problem. Project technology. Relevance of the project. Application of presentations. Content. Preface. Polyhedra inscribed in a sphere. Prism. We will answer verbally. A sphere is described around a triangular prism, the center of which lies outside the prism. Combination of sphere and prism. Measurements of a rectangular parallelepiped. A sphere of radius 5 cm is described around a regular hexagonal prism. Pyramid. A sphere can be described around any triangular pyramid. Combination of sphere and pyramid. The base of a triangular pyramid is a right triangle. Let's construct an axial section. Polyhedra described around a ball. - “Geometry problems” 11th grade.ppt

Plane equation

Linear algebra and analytical geometry. Topic: Plane. Plane. CONCLUSIONS: 1) The plane is a surface of the first order. Study of the general plane equation. Equation (3) is called the plane equation in segments. ?1: by+cz = 0 (intersection with the oyz plane) ?2: ax+by = 0 (intersection with the oxy plane). A) the plane cuts off segments a and b on the ox and oy axes, respectively, and is parallel to the oz axis; A) the plane cuts off segment a on the ox axis and is parallel to the oy and oz axes (i.e., parallel to the oyz plane); Comment. Let it be a plane? doesn't go through O(0;0;0). 2. Other forms of writing the plane equation. - Plane equation.pps

Planes in space

Analytic geometry. Part 2 Geometry in space. Analytical geometry in space. Equations of the plane. 1. Equation of a plane using a point and a normal vector. Given: a point and a normal vector Equation of a plane: Let a point Then. 2. General equation of the plane. An equation of the form is called a general equation of the plane. The coefficients A, B, C in the equation determine the coordinates of the normal vector: Theorem. 5. Coefficients A=B=0 (Fig. 5) 6. Coefficients A=C=0 (Fig. 6) 7. Coefficients B=C=0 (Fig. 7). 8. Coefficients A=B=D=0 9. Coefficients A=C=D=0 10. Coefficients B=C=D=0. -

The school geometry course consists of two parts:

Basic Concepts

Along with points, straight lines, planes, geometric bodies are considered in stereometry, their properties are studied, their areas are calculated

Volumetric geometric bodies

Points are designated by capital Latin letters A, B, C, D, E, K,...

Stereometry is widely used in construction

Stereometry is used in architecture

Stereometry is used in mechanical engineering

Stereometry is used in geodesy

It is clear that in each plane there are some points of space, but not all points of space lie in the same plane.

Axioms of stereometry

Some corollaries from the axioms

PLANIMETRY STEREOMETRY grades 7-9 classes GEOMETRY on the plane GEOMETRY in space “planimetry” is a name of mixed origin: from the Greek. metreo – to measure and lat. planum – flat surface (plane) “stereometry” – from Greek. stereos – spatial (stereon – volume). School course GEOMETRY

Studying STEREOMETRY at school We will conduct a systematic examination of the properties of geometric bodies in space. Let's learn various ways to calculate practically important geometric quantities. At the same time, we will develop spatial imagination and logical thinking

GEOMETRY arose from the practical problems of people; GEOMETRY underlies all technology and most inventions of mankind; GEOMETRY is needed GEOMETRY arose from the practical problems of people; GEOMETRY underlies all technology and most inventions of mankind; GEOMETRY is needed by a technician, engineer, worker, architect, fashion designer... technician, engineer, worker, architect, fashion designer... We know that

Intuitive, vivid spatial imagination, combined with strict logic of thinking, is the key to studying stereometry. CONCLUSION: When studying stereometry, we will use drawings, drawings: they will help us understand, imagine, illustrate the content of a particular fact. Therefore, before you begin to understand the essence of an axiom, definition, proof of a theorem, or solution of a geometric problem, try to visualize, imagine, and draw the figures in question. “My pencil can be even wittier than my head,” admitted the great mathematician Leonhard Euler ().

1. Any three points lie in the same plane. 2. Any four points lie in the same plane. 3. Any four points do not lie in the same plane. 4. A plane passes through any three points, and only one. 5.If a straight line intersects 2 sides of a triangle, then it lies in the plane of the triangle. 6.If a line passes through the vertex of a triangle, then it lies in the plane of the triangle. 7.If the lines do not intersect, then they are parallel. 8.If the planes do not intersect, then they are parallel. In stereometry, we will consider situations that specify different locations in space of the main figures relative to each other. Determine: is the judgment correct? NOT REALLY

Axioms of stereometry The word “axiom” is of Greek origin and in translation means the true, initial position of the theory. The system of axioms of stereometry gives a description of the properties of space and its main elements. The concepts of “point”, “straight line”, “plane”, “distance” are accepted without definitions: their description and properties are contained in the axioms

CONSEQUENCES FROM AXIOM T-1 Through any straight line and a point not belonging to it, one can draw a plane, and only one. m m A B Given: M m Since M is m, then points A, B and M do not belong to the same line. Along A-1, only one plane passes through points A, B and M (ABM). Let us denote it. Line m has two common points with it, points A and B, therefore, according to axiom A-2, this line lies in the plane. Thus, the plane passes through line m and point M and is the desired one. Let us prove that there is no other plane passing through the line m and the point M. Suppose that there is another plane passing through the line m and the point M. Then the planes and pass through the points A, B and M, which do not belong to the same line, and therefore coincide. Therefore, the plane is unique. The theorem is proven. Proof Let points A, B m.
CONSEQUENCES FROM AXIOM T-2 Through any two intersecting lines a plane can be drawn, and only one. N m m n Given: m n = M Proof Let us mark an arbitrary point N on the line m, different from M. Consider the plane =(n, N). Since M and N, then according to A-2 m. This means that both straight lines m, n lie in the plane and, therefore, is the desired one. Let us prove the uniqueness of the plane. Let us assume that there is another plane, different from the plane and passing through the lines m and n. Since the plane passes through the line n and the point N that does not belong to it, then along T-1 it coincides with the plane. The uniqueness of the plane has been proven. The theorem is proven

Like planimetry, stereometry is based on certain axioms, on the basis of which theorems will be proved and problems solved in the future. Axioms, as you know, do not require proof. If you skip this topic, then further study of stereometry will not make any sense. Solutions will become unclear, the student will lag behind his peers, and academic performance will decline in many ways. Therefore, it is worth studying this presentation thoroughly. This can be done in a classroom with a teacher, or at home. Having missed this topic, further solutions in subsequent presentations will not be clear, because they refer to the axioms in this lesson.

The presentation consists of 14 slides, the first of which recalls the definition of the concept of an axiom. Next, it is clarified what is an axiom in stereometry. The first axiom in this section says that only one plane can be drawn through three points. This is a very important statement. Schoolchildren should have a good understanding of this and understand that an infinite number of planes can be drawn through one or two points. An image of a plane drawn through three points is shown on the same slide.

The second axiom states that if some points of an arbitrary line (minimum 2) lie on a plane, then all an infinite number of points also lie on this plane. You can also verify this simply. However, it cannot be proven. That statement is an axiom. If students do not understand or do not understand a particular axiom, you can ask them to prove the opposite in a practical way. That is, give at least one example that will refute the statement. Thanks to this, they will be able to develop mathematical and spatial thinking.

The next axiom, A3, talks about the intersection of two planes about the common straight line that they have. Planes are depicted through parallelograms. There are also other ways to designate them, but this is the most common in many textbooks, including school ones.

The next slide shows images of the three axioms. It is advisable to redraw all these drawings in notebooks in order to better remember and understand. In this way, you can remember the axioms better. So, three main statements were considered, to which students will return repeatedly. It is advisable to know their wording and be able to use them correctly, and also reproduce them if necessary.

Next, the presentation suggests considering a problem in which a body such as a tetrahedron is studied. The schoolchildren were previously familiar with this figure, and most likely had dealings with it. In order for the teacher to understand whether students can cope with spatial thinking, it is proposed to determine some planes, intersection points, etc. against the background of this figure. If some people have difficulties, then they should be given similar examples at home so that they can better understand the essence.

After this problem there is another one. To solve it, you need to remember all the axioms you have learned and learn to use them. If there is time left from the lesson, it is worth reviewing as many practical problems as possible with the class.

With the help of the presentation “Axioms of Stereometry”, a young teacher can teach an interesting lesson and attract the attention of students. Thanks to optical perception, schoolchildren will be able to better assimilate and understand the material. When writing a plan for notes, which young teachers do without fail, a presentation will also come in handy. It will help you structure the lesson correctly and not miss a single axiom, not a single important explanation or remark.

The examples given in the presentation will also be useful when teaching the lesson.