Similar figures are the figures identical in a form, but different in the size. Triangles are similar if their corners are equal, and the parties are proportional each other. There are also three signs allowing to define similarity without observance of all conditions. Sign the first – at similar triangles two corners of one are equal to two corners of another. The second sign of similarity of triangles - two parties of one are proportional to two parties of another, and corners between these parties at them are equal. The third sign of similarity are a proportionality of three parties to one three parties of another.
It is required to you
- - handle;
- - note paper.
Instruction
1. The coefficient of similarity expresses proportionality, this relation of lengths of the parties of one triangle to the skhodstvenny parties of another: k = AB/A’B’ = BC/B’C’ = AC/A’C’. The Skhodstvenny parties are in triangles opposite to equal corners. Coefficient of similarity search different is possible in the ways.
2. For example, in a task similar triangles are given and lengths of their parties are specified. It is required to find similarity coefficient. As triangles are similar on a condition, find their skhodstvenny parties. For this purpose write down lengths of the parties of one and others on increase. Find the relation of the skhodstvenny parties which will be similarity coefficient.
3. You can calculate coefficient of similarity of triangles if their areas are known to you. One of properties of similar triangles says that the relation of their areas equals to a similarity coefficient square. Divide values of the areas of similar triangles one into another and take a square root from result.
4. The relations of perimeters, lengths of medians, the mid-perpendiculars constructed to the skhodstvenny parties are equal to similarity coefficient. If to divide length of the bisectors or heights which are carried out from identical corners you also receive similarity coefficient. Use this property for finding of coefficient if in a statement of the problem these sizes are given.
5. According to the theorem of sine for any triangle of the relation of the parties to sine of opposite corners are equal to diameter of the circle described around it. From this follows that at similar triangles the relation of radiuses or diameters of circumscribed circles is equal to similarity coefficient. If in a task radiuses of these circles are known, or they can be found from the areas of circles, find similarity coefficient this way.
6. Use a similar way for finding of coefficient if you have circles entered in similar triangles with the known radiuses.