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Tasks for solid geometry

task 1

The figures show the sweep of the pyramid ABCD dividing point it is necessary to identify these letters so as to obtain a triangular pyramid.
Without the help of models of these tasks are useful for everyone. They develop imagination and logical thinking. Such problems can be solved without the help of knowledge. If the solution does not work, then scan the model suggests a solution. The aforementioned main modifications pyramid, but opportunities are endless transformations. The simplicity of the device and versatility action opens an interesting and complex world of geometry, so the pyramid need to have on the table the student in the classroom

геометрии.

Задачи по стереометрии

task 2

The figure shows the sweep of the pyramid ABCDE, where E is the point of the quad.
Separating point it is necessary to designate these letters so as to obtain foursquare pyramid.

Задачи по стереометрии

task 3
The figure shows a hexagonal pyramid, which consists of 18 items.
How can one get moving one element of a 6-coal pyramid 7-coal?
Prototype problem exists for the 4-coal, coal-8, etc. pyramids.

Solving problems on solid geometry

vopros

Before proceeding to the solution of problems, it should be noted that the models are indispensable for drawings prostarnstvennyh bodies on the background of the board. Hold the model appropriately at the board, and you can draw yourself as visible. To solve the problems relating to the triangular pyramid is necessary to use a quadrangular pyramid, which is due to the fact that in various modified triangular pyramid, makes it possible to demonstrate specific tasks.

Task 1.
Have a triangular pyramid with equal edges whose base is a right triangle. The height of the pyramid passes through the center of the circle. Center of the circle in the middle of the hypotenuse.

Objective 2.
Ribs have a pyramid with a, b, c, which are mutually perpendicular. We need to find the volume of a pyramid. The task gets a simple solution if the inverted pyramid. As you can see, has turned the pyramid whose base is a right triangle, where the known length of the legs of the base (the base is a right-angled triangle) and the length of the altitude.
Thus, the volume of the pyramid is a, b, c ,. With our excellent fashion models demonstrated the possibility of obtaining drawings of spatial geometric bodies on the board.

Task 3.
Prove that in any pyramid can vpysat sphere. This is one of the most difficult problems in solid geometry. Wired section which passes through the dihedral angle bisector BACD. They intersect in a straight line AH. If you hold the third section, then it intersects with the second section in a straight line CM. The point of intersection CH and CM is the center of the inscribed sphere, as equidistant from all sides.

Demonstration and solving interesting problems

vopros Если цифру 1 подвесим к отметке 10 левого плеча, 2 к 1, то, подвесив 3 к отметке 4, получим равновесие с использованием последующих чисел 1, 2, 3.
Исходя из этого решения, предлагаем самостоятельно решить подобную задачу: все числа от 1 до 10 в порядке возрастания по одному подвесить на разных плечах весов и получить равновесие.
Обобщенная задача для n-натуральных чисел от автора С. В. Мовсисяна

Задача:
Дано. На математических весах шкала от 1 до n, подвесить все числа от 1 до n, так, чтобы получилось равновесие. Сколько вариантов ?

Решившего задачу ждет приз от автор в размере 50 тыс рублей.