Geometrical constructions - an important part of the program of training. They develop imagination, logic and spatial thinking. The majority of problems on constructions have to be solved only with the help of a ruler, compasses and pencil. It allows to fix perception of dependences between parameters of geometrical objects. Some of them are simple and natural, and some are not visible obviously. So to construct diagonals of a square or the isosceles triangle does not make special work, and how to divide a circle into 12 parts, it is necessary to think a little.
It is required to you
- Ruler, compasses, pencil.
Instruction
1. Draw a circle, or find the radius of the existing circle. If the circle is not set, then just draw it, having established convenient distance between compasses legs. Do not change this distance after the end of drawing of a circle. If it is necessary to divide already existing circle, it is necessary to determine its radius at first. For this purpose draw the piece crossing a circle in two points of A and B. Using compasses and a ruler, construct a perpendicular to a piece [A; B], dividing it into two equal parts. It will cross a circle in points of C and D. Construct a similar perpendicular to a piece [C; D]. Let it cross a circle in points of E and F. Crossing of pieces [E; F] and [C; D] will be the center of a circle. Put a compasses needle in any point of a circle and move other his leg so that it was established in a point of intersection of pieces [E; F] and [C; D]. Radius of a circle is found.
2. Divide a circle into six parts. Establish a compasses needle in any point of a circle. Draw two arches crossing a circle in two points. The distance between legs of compasses has to be equal to circle radius. In other words, it has to be such as it was established on the previous step. Move a compasses leg with a needle to a point of intersection of one and arches with a circle. Again draw two arches crossing a circle. Move a compasses leg to the following points of intersection of arches with a circle and build arches, you will not find yet six points which divide a circle into six equal parts. Let it will be points of A, B, C, D, E, F.
3. Construct the correct hexagon, inscribed in a circle. For this purpose consistently connect points of A-B-C-D-E-F-A.
4. Divide a circle into twelve parts. Construct perpendiculars to pieces [A; B], [B; C], [C; D], dividing them into two equal parts. Let A', B', C', D', E', F will be points of intersection of these perpendiculars with a circle'. Points of A, A', B, C', C, E', D, B', E, D', F divide a circle into twelve equal parts.