Section in a cube. “Section of a cube by a plane and their practical application in problems”

Tasks on Constructing sections of a cubeD1
C1
E
A1
B1
D
A
F
B
WITH

Verification work.

1 option
Option 2
1. tetrahedron
1. parallelepiped
2. Properties of a parallelepiped

A cutting plane of a cube is any plane on both sides of which there are points of a given cube.

Secant
the plane intersects the faces of the cube along
segments.
A polygon whose sides are
These segments are called a section of the cube.
The sections of a cube can be triangles,
quadrilaterals, pentagons and
hexagons.
When constructing sections, one should take into account that
fact that if a cutting plane intersects two
opposite faces along some segments, then
these segments are parallel. (Explain why).

B1
C1
D1
A1
M
K
IMPORTANT!
B
WITH
D
If the cutting plane intersects
opposite edges, then it
K DCC1
intersects them in parallel
M BCC1
segments.

three given points that are the midpoints of the edges. Find the perimeter of the section if the edge

Construct a section of the cube with a plane passing through
three given points that are the midpoints of the edges.
Find the perimeter of the section if the edge of the cube is equal to a.
D1
N
K
A1
D
A
C1
B1
M
WITH
B

Construct a section of the cube with a plane passing through three given points, which are its vertices. Find the perimeter of the section if the edge of the cube

Construct a section of the cube with a plane passing through
three given points that are its vertices. Find
the perimeter of the section if the edge of the cube is equal to a.
D1
C1
A1
B1
D
A
WITH
B

D1
C1
A1
M
B1
D
A
WITH
B

Construct a section of the cube with a plane passing through three given points. Find the perimeter of the section if the edge of the cube is equal to a.

D1
C1
A1
B1
N
D
A
WITH
B

Construct a section of the cube with a plane passing through three given points, which are the midpoints of its edges.

C1
D1
B1
A1
K
D
WITH
N
E
A
M
B

Lesson Objectives

Formation of students' skills in solving problems involving the construction of sections.
Formation and development of spatial imagination in students.
Development of graphic culture and mathematical speech.
Developing the ability to work individually and in a team.

Lesson type: lesson in the formation and improvement of knowledge.

Forms of organizing educational activities: group, individual, collective.

Lesson technical support: computer, multimedia projector, screen, set of geometric bodies (cube, parallelepiped, tetrahedron).

DURING THE CLASSES

1. Organizational moment

The class is divided into 3 groups of 5-6 people. On each table there are individual and group tasks for constructing a section, a set of bodies. Introducing students to the topic and objectives of the lesson.

2. Update background knowledge

Poll theory:

– Axioms of stereometry.
– The concept of parallel lines in space.
– Theorem on parallel lines.
– Parallelism of three straight lines.
– The relative position of a straight line and a plane in space.
– A sign of parallelism between a line and a plane.
– Determination of parallelism of planes.
– Sign of parallelism of two planes.
– Properties of parallel planes.
– Tetrahedron. Parallelepiped. Properties of a parallelepiped.

3. Learning new material

Teacher's word: When solving many stereometric problems, a section of a polyhedron by a plane is used. Let us call a secant plane of a polyhedron any plane on both sides of which there are points of the given polyhedron.
The cutting plane intersects the faces along segments. The polygon whose sides are these segments is called a section of the polyhedron.
Using Figures 38-39, let's find out: How many sides can a cross-section of a tetrahedron and a parallelepiped have?

Students analyze the pictures and draw conclusions. Teacher corrects students' answers, pointing out the fact that if a cutting plane intersects two opposite faces of a parallelepiped along some segments, then these segments are parallel.

Analysis solving problems 1, 2, 3 given in the textbook (oral group work).

4. Consolidation of the studied material(by groups)

1 group: explain how to construct a section of a tetrahedron with a plane passing through given points M, N, K and in problems 1-3 find the perimeter of the section if M, N, K are the midpoints of the edges and each edge of the tetrahedron is equal A.

Group 2: explain how to construct a section of a cube with a plane passing through three given points, which are either the vertices of the cube or the midpoints of its edges (the three given points are highlighted in the figures); in problems 1-4 and 6, find the perimeter of the section if the edge of the cube is equal to A. in problem 5 prove that AE = A/3

Group 3: construct a cross section of a parallelepiped ABCDA 1 B 1 C 1 D 1 plane passing through the points:

The group defends all completed tasks at the board using slides.

5. Independent work № 85, № 105.

6. Summing up the lesson

Assessing students' work in class.

7. Homework: individual cards.

Tasks on Constructing sections of a cubeD1
C1
E
A1
B1
D
A
F
B
WITH

Verification work.

1 option
Option 2
1. tetrahedron
1. parallelepiped
2. Properties of a parallelepiped

A cutting plane of a cube is any plane on both sides of which there are points of a given cube.

B1
C1
D1
A1
M
K
IMPORTANT!
B
WITH
D
If the cutting plane intersects
opposite edges, then it
K DCC1
intersects them in parallel
M BCC1
segments.

three given points that are the midpoints of the edges. Find the perimeter of the section if the edge

Construct a section of the cube with a plane passing through three given points, which are its vertices. Find the perimeter of the section if the edge of the cube

Construct a section of the cube with a plane passing through
three given points that are its vertices. Find
the perimeter of the section if the edge of the cube is equal to a.
D1
C1
A1
B1
D
A
WITH
B

D1
C1
A1
M
B1
D
A
WITH
B

Construct a section of the cube with a plane passing through three given points. Find the perimeter of the section if the edge of the cube is equal to a.

D1
C1
A1
B1
N
D
A
WITH
B

Construct a section of the cube with a plane passing through three given points, which are the midpoints of its edges.

C1
D1
B1
A1
K
D
WITH
N
E
A
M
B

Lesson type: Combined lesson.

Goals and objectives:

educational – formation and development of spatial concepts in students; developing skills in solving problems involving constructing sections of the simplest polyhedra;
educational - cultivate the will and perseverance to achieve final results when constructing sections of the simplest polyhedra; Foster a love and interest in learning mathematics.
developing – student development logical thinking, spatial representations, development of self-control skills.

Equipment: computers with a specially developed program, handouts in the form ready-made drawings with tasks, solids of polyhedra, individual cards with homework.

Lesson structure:

State the topic and purpose of the lesson (2 min).
Instructions on how to complete tasks on a computer (2 min).
Updating students' basic knowledge and skills (4 min).
Self-test (3 min).
Solving problems with an explanation of the solution by the teacher (15 min).
Independent work with self-test (10 min).
Setting homework (2 min).
Summing up (2 min).

During the classes

1. Communicating the topic and purpose of the lesson

After checking the class’s readiness for the lesson, the teacher reports that today there is a lesson on the topic “Constructing sections of polyhedra”; problems will be considered on constructing sections of some simple polyhedra with planes passing through three points belonging to the edges of the polyhedra. The lesson will be taught using a computer presentation made in Power Point.

2. Safety instructions when working in a computer lab

Teacher. I draw your attention to the fact that you are starting to work in a computer class, and you must follow the rules of conduct and work at the computer. Secure retractable tabletops and ensure proper fit.

3. Updating the basic knowledge and skills of students

Teacher. To solve many geometric problems related to polyhedra, it is useful to be able to construct their sections in a drawing using different planes, find the point of intersection of a given line with a given plane, and find the line of intersection of two given planes. In previous lessons, we looked at sections of polyhedra by planes parallel to the edges and faces of the polyhedra. In this lesson we will look at problems involving constructing sections with a plane passing through three points located on the edges of polyhedra. To do this, consider the simplest polyhedra. What are these polyhedra? (Models of a cube, tetrahedron, regular quadrangular pyramid, straight triangular prism).

Students must determine the type of polyhedron.

Teacher. Let's see how they look on the monitor screen. We move from image to image by pressing the left mouse button.

Images of the named polyhedra appear on the screen one after another.

Teacher. Let us remember what is called a section of a polyhedron.

Student. A polygon whose sides are segments belonging to the faces of the polyhedron, with ends on the edges of the polyhedron, obtained by intersecting the polyhedron with an arbitrary cutting plane.

Teacher. What polygons can be sections of these polyhedra.

Student. Sections of a cube: three - hexagons. Sections of a tetrahedron: triangles, quadrangles. Sections of a quadrangular pyramid and a triangular prism: three - pentagons.

4. Self-testing

Teacher. In accordance with the concept of sections of polyhedra, knowledge of the axioms of stereometry and the relative position of lines and planes in space, you are asked to answer the test questions. The computer will appreciate you. Maximum score 3 points – for 3 correct answers. On each slide you must click the button with the number of the correct answer. You work in pairs, so each of you will receive the same computer-specified number of points. Click the next slide indicator. You have 3 minutes to complete the task.

I. Which figure shows a section of a cube by a plane ABC?

II. Which figure shows a cross section of a pyramid with a plane passing through the diagonal of the base? BD parallel to the edge S.A.?

III. Which figure shows a cross section of a tetrahedron passing through a point M parallel to the plane ABS?

5. Solving problems with an explanation of the solution by the teacher

Teacher. Let's move on directly to solving problems. Click the next slide indicator.

Problem 1 This task Let's look at it orally with a step-by-step demonstration of the construction on the monitor screen. The transition is carried out by clicking the mouse.

Given a cube ABCDAA 1 B 1 C 1 D 1 . On his edge BB 1 given point M. Find the point of intersection of a line C 1 M with the plane of the cube face ABCD.

Consider the image of a cube ABCDAA 1 B 1 C 1 D 1 with a dot M on the edge BB 1 Points M And WITH 1 belong to the plane BB 1 WITH 1 What can be said about the straight line C 1 M ?

Student. Straight C 1 M belongs to the plane BB 1 WITH 1

Teacher. Searched point X belongs to the line C 1 M, and therefore planes BB 1 WITH 1 . What's it like mutual arrangement planes BB 1 WITH 1 and ABC?

Student. These planes intersect in a straight line B.C..

Teacher. That means everything common points planes BB 1 WITH 1 and ABC belong to the line B.C.. Searched point X must simultaneously belong to the planes of two faces: ABCD And BB 1 C 1 C; from this it follows that point X must lie on the line of their intersection, i.e. on the straight line Sun. This means that point X must lie on two straight lines simultaneously: WITH 1 M And Sun and, therefore, is their point of intersection. Construction the desired point look at it on the monitor screen. You will see the construction sequence by pressing the left mouse button: continue WITH 1 M And Sun to the intersection at the point X, which is the desired intersection point of the line WITH 1 M with face plane ABCD.

Teacher. To move to the next task, use the next slide indicator. Let's consider this problem with a brief description of the construction.

A) Construct a section of a cube with a plane passing through the points A 1 , MD 1 C 1 and NDD 1 and b) Find the line of intersection of the cutting plane with the plane of the lower base of the cube.

Solution. I. The cutting plane has a face A 1 B 1 C 1 D 1 two common points A 1 and M and, therefore, intersects with it along a straight line passing through these points. Connecting the dots A 1 and M straight line segment, we find the line of intersection of the plane of the future section and the plane top edge. We will record this fact in the following way: A 1 M. Press the left mouse button, pressing again will construct this straight line.

Similarly, we find the lines of intersection of the cutting plane with the faces AA 1 D 1 D And DD 1 WITH 1 WITH. By clicking the mouse button, you will see a brief recording and construction progress.

Thus, A 1 NM? the desired section.

Let's move on to the second part of the problem. Let's find the line of intersection of the cutting plane with the plane of the lower base of the cube.

II. The cutting plane intersects with the plane of the base of the cube in a straight line. To depict this line, it is enough to find two points belonging to this line, i.e. common points of the cutting plane and the face plane ABCD. Relying on previous task such points will be: point X=. Press the key, you will see a short recording and construction. And period Y, what do you guys think, how to get it?

Student. Y =

Teacher. Let's look at its construction on the screen. Click the mouse button. Connecting the dots X And Y(Record X-Y), we obtain the desired straight line - the line of intersection of the cutting plane with the plane of the lower base of the cube. Press the left mouse button - short recording and construction.

Problem 3 Construct a section of the cube with a plane passing through the points:

Also, by pressing the mouse button, you will see the construction progress and a short recording on the monitor screen. Based on the concept of a section, it is enough for us to find two points in the plane of each face to construct the line of intersection of the cutting plane and the plane of each face of the cube. Points M And N belong to the plane A 1 IN 1 WITH 1 . By connecting them, we get the line of intersection of the cutting plane and the plane of the upper face of the cube (press the mouse button). Let's continue the straight lines MN And D 1 C 1 before the intersection. Let's get a point X, belonging to both the plane A 1 IN 1 WITH 1 and plane DD 1 C 1 (mouse click). Points N And TO belong to the plane BB 1 WITH 1 . By connecting them, we get the line of intersection of the cutting plane and the face BB 1 WITH 1 WITH. (Mouse click). Connecting the dots X And TO, and continue straight HC to the intersection with the line DC. Let's get a point R and segment KR – line of intersection of the cutting plane and the face DD 1 C 1 C. (Mouse click). Continuing straight KR And DD 1 before intersection, we get a point Y, belonging to the plane AA 1 D 1 . (Mouse click). In the plane of this face we need one more point, which we obtain as a result of the intersection of lines MN And A 1 D 1 . This is the point . (Mouse click). Connecting the dots Y And Z, we get And . (Mouse click). Connecting Q And R, R And M, will we get it? the desired section.

Brief description of the construction:

2) ;

6) ;

7) ;

13) ? the desired section.

Lesson topic: Tasks on constructing sections.

The purpose of the lesson:

Develop skills in solving problems involving constructing sections of a tetrahedron and parallelogram.

During the classes

I. Organizational moment.

II. Checking homework

Answers to questions 14, 15.

14. Is there a tetrahedron with five right angles on its faces?

(Answer: no, because there are only 4 faces, they are triangles, and a triangle with two right angles does not exist.)

15. Is there a parallelepiped that has: a) only one face - a rectangle;

b) only two adjacent rhombus faces; c) all the corners of the faces are sharp; d) all angles of the faces are right; e) the number of all sharp edges is not equal to the number of all obtuse angles of the faces?

(Answer: a) no (opposite sides are equal); b) no (for the same reason); c) no (such parallelograms do not exist); d) yes ( cuboid); e) no (each face has two sharp and two obtuse angles, or all straight lines).

III. Learning new material

Theoretical part. Practical part. Theoretical part.

To solve many geometric problems associated with a tetrahedron and a parallelepiped, it is useful to be able to draw their sections in different planes in a drawing. By section we mean any plane (let's call it a cutting plane), on both sides of which there are points of a given figure (that is, a tetrahedron or parallelepiped). The cutting plane intersects the tetrahedron (parallelepiped) along segments. The polygon that will be formed by these segments is the cross section of the figure. Since a tetrahedron has four faces, its cross-section can be triangles and quadrangles. The parallelepiped has six faces. Its cross-section can be triangles, quadrangles, pentagons, hexagons.

When constructing a section of a parallelepiped, we take into account the fact that if a cutting plane intersects two opposite faces along some segments, then these segments are parallel (property 1, paragraph 11: If two parallel planes are crossed by the third, then the lines of their intersection are parallel).

To construct a section, it is enough to construct the points of intersection of the cutting plane with the edges of the tetrahedron (parallelepiped), and then draw segments connecting each two constructed points lying on the same face.

Can a tetrahedron be cut by a plane into the quadrilateral shown in the figure?

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2.2. Construct a section of a cube with a plane passing through the points E, F, G, lying on the edges of the cube.

E, F, G,

let's make a direct E.F. and denote P its point of intersection with AD.

Let's denote Q point of intersection of lines PG And AB.

Let's connect the dots E And Q, F And G.

The resulting trapezoid EFGQ will be the desired section.

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2.4. Construct a section of a cube with a plane passing through the points E, F, lying on the edges of the cube and the vertex B.

Solution. To construct a section of a cube passing through points E, F and the top B,

Let's connect the points with segments E And B, F And B.

Through dots E And F let's draw parallel lines B.F. And BE, respectively.

The resulting parallelogram BFGE will be the desired section.

2.5. Construct a section of a cube with a plane passing through the points E, F, G, lying on the edges of the cube.

Solution. To construct a section of a cube passing through points E, F, G,

let's make a direct E.F. and denote P its point of intersection with AD.

Let's denote Q,R line intersection points PG With AB And DC.

Let's denote S intersection point FR c SS 1.

Let's connect the dots E And Q, G And S.

The resulting pentagon EFSGQ will be the desired section.

2.6. Construct a section of a cube with a plane passing through the points E, F, G, lying on the edges of the cube.

Solution. To construct a section of a cube passing through points E, F, G,

let's find a point P intersection of a straight line E.F. and face plane ABCD.

Let's denote Q, R line intersection points PG With AB And CD.

Let's make a direct RF and denote S, T its points of intersection with CC 1 and DD 1.

Let's make a direct T.E. and denote U its point of intersection with A 1D 1.

Let's connect the dots E And Q, G And S, F and U.

The resulting hexagon EUFSGQ will be the desired section.

2.7. Construct a cross section of a tetrahedron ABCD AD and passing through the points E, F.

Solution. Let's connect the dots E And F. Through the pointF let's draw a straight lineFG, parallelA.D.

Let's connect the dots G And E.

The resulting triangle EFG will be the desired section.

2.8. Construct a cross section of a tetrahedron ABCD flat, parallel to the edge CD and passing through the points E, F .

Solution. Through dots E And F let's draw straight lines E.G. And FH, parallel CD.

Let's connect the dots G And F, E And H.

The resulting triangle EFG will be the desired section.

2.9. Construct a cross section of a tetrahedron ABCD plane passing through the points E, F, G.

Solution. To construct a section of a tetrahedron passing through points E, F, G,

let's make a direct E.F. and denote P its point of intersection with BD.

Let's denote Q point of intersection of lines PG And CD.

Let's connect the dots F And Q, E And G.

The resulting quadrilateral EFQG will be the desired section.

IV. Lesson summary.

V. Homework p.14, p.27 No. 000 – option 1, 2.