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# Geometric educational instructional manual "Triangular Pyramid" and main transformations

Before starting working with the models it is necessary to be aware that for manipulations and transformations it is needed to work consistently only with each side separately. For the implementation of each of the following transformations it is needed to close consistently each of the sides up to the stop.

Getting the geometric figures out of triangular pyramid model

To implement the conversion it is necessary to close consistently each side of a triangular pyramid ABCD up to the stop and get the correct triangular pyramid (tetrahedron).

1. Romb–is obtained from the original pyramid through BD side stretching/pulling

2.
Quadrat- is obtained from therhomb through AC side stretching/pulling and DB shortening

3. Trapezoiis obtained through stretching/pulling of the sides AB, as a result the diagonal AC and BD are stretched themselves
d
4. Parallelogramisobtainedthroughstretching/pullingofthesidesDCandthediagonalAC. Here it is possible to demonstrate operation on vectors (addition and subtraction).

5.  Rectangle–is obtained stretching/pulling BD and shortening AC

6.  Triangle

а) Close each side of the model sequentially up to the stop - the starting point is obtained
б) All sides of the base of pyramid ABC should be stretched/pulled until the edges DA, DB, DC will fit the four centers lie in a planeof the triangle ABC

7.  Test of equality of triangles

AD must be disconnected at point D, and the AC and BD have to be stretched/pulled until the top of D does not coincide with the vertex A.
Watch the video "A triangular pyramid" with 0:56 seconds

8.  The characteristics of an isosceles triangle

Starting position –is tetrahedron. In the beginning it is necessary to obtain a rhomb, and thenstretch/pull the rods CB, AB and DB.

9.  Pyramid

The pyramid can be obtained from the triangle (point 6)

The top/vertex D starts to move away from the tops/vertexes A, B and C a triangular pyramid is being formed.

9.1 The correct triangular pyramid DА=DВ=DС и АВ=ВС=АС.

9.2 The projection of the triangle DAC in the plane of ABC is obtained by stretching DA and DC until DB does not become perpendicular to the plane of ABC.

9.3 The theorem on three perpendiculars/verticals.

It is necessary to stretch/pull the BC and AC until the side of the CA will not be perpendicular to AB. Studying the theorem of three perpendiculars, the pupils are asked to convert the four triangles of the pyramid ABCD into the right triangles which is not that simple. When repeated attempts of pupils are unsuccessful, the teacher, being inferior in resourcefulness to pupils, but superior in knowledge, shows the miracle about three perpendiculars.

9.4

1.1   The planes (DCA) and (DBC) are perpendicular to the plane ABC. From this it follows that the line of intersection of the planes DC is perpendicular to the plane of ABC.

10A solution of the problem

Task. Given: a pyramid with ribs/links a, b, c, which are mutually perpendicular.Findthevolumeof the pyramid.