The results of the preliminary analysis of tasks must be somehow recorded, written down. The verbal, descriptive form of writing, which is given above, is of course not very convenient. We need a more convenient, more compact and at the same time quite visual form of recording the analysis of the problem. Such a record is a schematic record of the task. For a schematic record of a geometric problem, it is useful to use a drawing of the figure that is considered in the problem.

The technique of constructing a drawing from geometry

When building such a drawing, a number of requirements must be met. Here are some of them:

1. The drawing should be a schematic drawing of the main object of the problem (a geometric figure, or a set of figures, or some part of these figures) with the designation using letters and other signs of all the elements of the figure and some of its characteristics. If any designations of the figure or its elements are indicated in the text of the problem, then these designations should also be in the drawing; if there are no designations in the problem, then you should use generally accepted designations or come up with the most convenient ones.

2. This drawing should be appropriate for the task. This means that if, for example, a triangle is named as the main object in the problem, and at the same time its form (rectangular, isosceles, etc.) is not indicated, then it is necessary to build some kind of versatile triangle.

3. When building a drawing, there is no need to maintain strictly any specific scale. However, it is advisable to observe some proportions in the construction of individual elements of the figure. If, according to the condition of the problem, the side AB of the triangle ABC is the largest, then this must be observed in the drawing. Or if the median of a triangle is specified, then the corresponding segment in the drawing should pass approximately through the middle of the side of the triangle, etc.

4. When building drawings of spatial figures, you must follow all the rules for drawing these figures. Where it is possible and expedient, it is better to build any plane sections of these figures.

Examples of construction in geometry

In addition to the drawing, a short record of all the conditions and requirements of the problem is also used for the schematic recording of geometric problems. In this short entry, using the designations accepted in the drawing, all the characteristics and relationships specified in the conditions of the problem are recorded. It is advisable to replace the name of the figures or its individual parts with a record of their definitions. For example, instead of writing: ABCD is a trapezoid, you can write: AD || CD. In the shorthand, you can use, where appropriate, standard mathematical signs (sign of membership, intersection, etc.)

Let us consider with examples how schematic records of geometric problems are constructed using drawings:

The diagonal of the trapezoid is perpendicular to its bases; the obtuse angle adjacent to the larger base is 1200, and the lateral side adjacent to it is 7cm; the larger base is 12cm. Find the midline of the trapezoid.

The main object of this task is the trapezoid. In this trapezoid, the diagonal is perpendicular to its bases. If you start drawing this trapezoid in the usual way, starting with the construction of the sides, then you can be mistaken. It is better to start construction from the indicated diagonal. The diagonal is perpendicular to both bases. It can be represented as follows: A diagonal is a vertical segment, from the ends of which two horizontal segments (the base of the trapezoid) depart, and in different directions. It is clear that the corners of the trapezoid of the vertices that this diagonal connects must both be obtuse. Indeed, in the problem it is given that the angle for a larger base is 1200. Now it is no longer difficult to complete this trapezoid. Using the designations accepted in the drawing, we write down all the conditions and requirements of the problem.