Construction of sections of a tetrahedron and parallelepiped Victoria Viktorovna Tkacheva, mathematics teacher at school No. 183 with in-depth study of the English language. St. Petersburg, 2011. Contents: 1. Goals and objectives 2. Introduction 3. The concept of a cutting plane 4. Definition of a section 5. Rules for constructing sections 6. Types of sections of a tetrahedron 7. Types of sections of a parallelepiped 8. Problem of constructing a section of a tetrahedron with an explanation 9. Problem of constructing a section of a tetrahedron with explanation 10. The task of constructing a cross-section of a tetrahedron using guiding questions 11. The second solution to the previous problem 12. The task of constructing a cross-section of a parallelepiped 13. The task of constructing a cross-section of a parallelepiped 14. Sources of information 15. Wishes for students Purpose of the work: Development of spatial concepts in students. Objectives: Introduce the rules for constructing sections. Develop skills in constructing sections of a tetrahedron and parallelepiped in various cases of specifying a cutting plane. To develop the ability to apply the rules for constructing sections when solving problems on the topics “Polyhedra”. To solve many geometric problems it is necessary to construct their sections using different planes. The cutting plane of a parallelepiped (tetrahedron) is any plane on both sides of which there are points of a given parallelepiped (tetrahedron). L The cutting plane intersects the faces of the tetrahedron (parallelepiped) along segments. L A polygon whose sides are these segments is called a section of a tetrahedron (parallelepiped). To construct a section, you need to construct the intersection points of the cutting plane with the edges and connect them with segments. In this case, it is necessary to take into account the following: 1. You can only connect two points lying in the plane of one face. 2. A cutting plane intersects parallel faces along parallel segments. 3. If only one point is marked in the face plane, belonging to the section plane, then an additional point must be constructed. To do this, it is necessary to find the intersection points of the already constructed lines with other lines lying on the same faces. What polygons can be obtained in a section? A tetrahedron has 4 faces. Its sections can produce: Triangles Quadrangles. A parallelepiped has 6 faces Triangles Pentagons. Its sections can produce: Quadrangles Hexagons Construct a section of the tetrahedron DABC with a plane passing through the points M,N,K D M AA 1. Let's draw a straight line through points M and K, because they lie on the same face (ADC). N K BB C C 2. Let's draw a straight line through points K and N, because they lie on the same face (CDB). 3. Using similar reasoning, we draw the straight line MN. 4. Triangle MNK – the required section. Construct a section of the tetrahedron with a plane passing through points E, F, K. 1. Draw KF. 2. We carry out FE. 3. Continue with EF, continue with AC. D F 4. EF AC =M 5. Carry out MK. E M C 6. MK AB=L A L K Rules B 7. Draw EL EFKL – the required section Construct a section of the tetrahedron with a plane passing through the points E, F, K. With which straight line a point lying at Which can Connect the resulting Which at once are bordered can we continue to get points that lie in the same connect? connect the resulting additional point? faces, name the section. extra point? D and E AC ELFK FSEK and point K, and FK F L C M A E K B Rules Second method Construct a section of a tetrahedron with a plane passing through points E, F, K. D F L C A E K B Rules First method O Method No. 1. Method number 2. Conclusion: regardless of the construction method, the sections are the same. Construct a section of a parallelepiped with a plane passing through points M, A, D. В1 D1 E A1 С1 В А 1. AD 2. MD 3. ME//AD, because (ABC)//(A1B1C1) 4. AE 5. AEMD – section. M D C Construct sections of a parallelepiped with a plane passing through points B1, M, N Rules B1 D1 C1 A1 P K B D A E N C O M 1. MN 3.MN ∩ BA=O 2. Continue 4. B1O MN,BA 5 . В1О ∩ А1А=К 6. КМ 7. Continue with MN and BD. 8. MN ∩ BD=E 9. B1E 10. B1E ∩ D1D=P, PN Sources of information 1. Geometry 10-11: textbook for general education. institutions / L.S. Atanasyan, V.F. Butuzov and others, M. Prosveshchenie 2. Problems for geometry lessons grades 7-11 / B.G. Ziv, St. Petersburg, NPO “Peace and Family”, ed. -in "Acacia". 3. Mathematics: A large reference book for schoolchildren and those entering universities / D.I.Averyanov, P.I.Altynov - M.: Bustard YOU HAVE LEARNED A LOT AND SEEN A LOT! SO GO GUYS: BE GOOD AND CREATE! THANK YOU FOR YOUR ATTENTION.
Purpose of the work:
Tasks:
To solve many geometric problems it is necessary to construct them sections different planes.
Cutting plane parallelepiped (tetrahedron) is any plane on both sides of which there are points of a given parallelepiped (tetrahedron).
Cutting plane intersects the faces of a tetrahedron (parallelepiped) along segments.
L
Polygon whose sides are these segments is called cross section tetrahedron (parallelepiped).
To construct a section, you need to construct the intersection points of the cutting plane with the edges and connect them with segments.
The following must be taken into account:
1. You can only connect two points lying
in the plane of one face.
2. A cutting plane intersects parallel faces along parallel segments.
3. If only one point is marked in the face plane, belonging to the section plane, then an additional point must be constructed. To do this, it is necessary to find the intersection points of the already constructed lines with other lines lying on the same faces.
What polygons can be obtained in a section?
A tetrahedron has 4 faces
The sections may look like:
The parallelepiped has 6 faces
In its sections
may turn out:
Construct a cross section of a tetrahedron DABC plane passing through the points M , N , K
points M and K, because they are lying
in one face (A DC).
2. Let's draw a straight line through points K and N, because they lie on the same face (C DB).
3. Using similar reasoning, we draw the straight line MN.
4. Triangle MNK –
the desired section.
passing through points E , F , K .
1. We carry out K F.
2. We carry out FE.
3. Continue with EF, continue with AC.
5. We carry out MK.
7. Conduct EL
EFKL – required
Construct a section of a tetrahedron by a plane,
passing through points E , F , K .
With F point
F and K, E and K
Construct a section of a tetrahedron by a plane,
passing through points E , F , K .
Method number 2.
Method number 1.
Conclusion: regardless of the construction method, the sections are the same.
Construct sections of a parallelepiped with a plane passing through points B 1, M, N
7. Let's continue with MN and BD.
2.Continue MN,BA
10. B 1 E ∩ D 1 D=P, PN
Construct a section of a parallelepiped by a plane,
passing through points M,A,D.
3. ME//AD, because (ABC)//(A 1 B 1 C 1)
5. AEMD – section.
YOU HAVE LEARNED A LOT
AND WE SAW A LOT!
SO GO GUYS:
BE DARY AND CREATE!
THANK YOU FOR YOUR ATTENTION.
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Lesson objectives:
Equipment: projector, interactive whiteboard, handouts.
Lesson type: lesson of learning new material.
Methods and techniques used in the lesson: visual, practical, problem-search, group, elements of research activity.
The teacher announces the topic and purpose of the lesson ( slide 1).
Teacher: While doing your homework, you had to find the meeting points of straight lines and planes, the trace of a cutting plane on the plane of the face of a polyhedron. Comment on what needs to be done for this.
(Students comment on homework ( slides 2-3).
Teacher: To move on to studying a new topic, let's review the theoretical material by answering the questions:
Teacher: Let's do a little research and answer the question: “What figure can be obtained in the section of a tetrahedron or parallelepiped by a plane?”
(Students, working in groups, look for the answer to the question posed.)
(After a few minutes they formulate their assumptions, and a demonstration begins slides 6–7.)
Teacher: Let's repeat the rules that need to be remembered when constructing sections of a polyhedron (students remember and formulate the necessary axioms, theorems, properties):
Teacher: Find errors in these drawings, justify your statement ( slides8-9).
Teacher: So, guys, we have prepared a theoretical basis for learning how to construct sections of polyhedra with a plane, in particular sections of a tetrahedron and parallelepiped. You will complete most of the tasks independently, working in groups, so each of you has worksheets with blank drawings of polyhedra on which you will build sections. If necessary, you can seek advice from a teacher or a senior in the group.
So, we present to your attention first task: (slide 10) construct a section of the tetrahedron with a plane passing through the given points M, N, K. (The section produces a triangle, check - slide 11.)
Teacher: Let's consider second task: Given the tetrahedron DABC. Construct a section of the tetrahedron by the plane MNK if M ∈DC, N∈AD, K∈AB. ( Slide 12)
(Solve the problem with the class, commenting on the construction.)
(Problem 3– independent work in groups ( slide 14). Examination - slide 15.)
Problem 4: Construct a section of the tetrahedron using the MNK plane, where M and N are the midpoints of the edges AB and BC ( slide 16). (Check for slide 17.)
Teacher: Let's move on to the next part of the lesson. Let's consider the problem of constructing sections of a parallelepiped by a plane. We found out that when a parallelepiped is sectioned by a plane, it can result in a triangle, quadrangle, pentagon or hexagon. The rules for constructing sections are the same. I suggest moving on to the next problem, which you will solve on your own.
(Demonstrated slide 18)
Problem 5
Construct a section of the parallelepiped ABCDA 1 B 1 C 1 D 1 by the plane MNK if M∈AA 1 , N ∈BB 1 , K∈CC 1 . (Check for slide 19).
Problem 6: (Slide 20) Construct a section of the parallelepiped ABCDA 1 B 1 C 1 D 1 by the PTO plane, if P, T, O belong to the edges AA 1, BB 1, CC 1, respectively.
(The solution is discussed, students construct a section on individual sheets and record the progress of construction ( slide 21).)
Task 7: (slide 22) Construct a section of the parallelepiped by the KMN plane if K ∈ A 1 D 1 , N ∈BC , M ∈ AB.
Solution: ( slide 23)
MPKFEN is the required section.
Creative tasks (cards according to options):
So, we got acquainted with the rules for constructing sections of a tetrahedron and parallelepiped, examined the types of sections, and solved the simplest problems for constructing sections. In the next lesson we will continue to study the topic and look at more complex problems.
Now let’s summarize the lesson by answering our traditional questions ( slide 24):
(Grading for the lesson.)
paragraph 14 105, 106. ( slide 25)
Additional task to 105: Find the ratio in which the plane MNK divides the edge AB if CN: ND = 2:1, BM = MD and point K is the midpoint of the median AL of triangle ABC.
(Finish the creative task.)
