The median is the piece connecting top of a triangle and the middle of the opposite party. Knowing lengths of all three parties of a triangle, it is possible to find its medians. In special cases of an isosceles and equilateral triangle, obviously, there is enough knowledge, respectively, two (not equal each other) and one party of a triangle. The median can also be found also according to other data.
It is required to you
- Lengths of the parties of a triangle, corners between the parties of a triangle
Instruction
1. Let's consider the most general case of a triangle of ABC with three parties not equal each other. Length of a median of AE of this triangle can be calculated on a formula: AE = sqrt (2*(AB^2)+2*(AC^2)-(BC^2))/2. Other medians are absolutely similarly. This formula is removed through Stewart's theorem, or through a triangle dostroyeniye to a parallelogram.
2. If the triangle of ABC - isosceles and AB = AC, then a median of AE is at the same time and height of this triangle. Therefore, the triangle of BEA will be rectangular. On Pythagorean theorem, AE = sqrt ((AB^2)-(BC^2)/4). From the general formula of length of a median of a triangle, for medians of BO and SP it is fair: BO = CP = sqrt (2 * (BC^2)+ (AB^2))/2.
3. If a triangle of ABC - equilateral, then, it is obvious that all its medians are equal each other. As the corner at top of an equilateral triangle is equal to 60 degrees, AE = to BO = to CP = to a*sqrt (3)/2 where a = AB = AC = BC is length of the party of an equilateral triangle.
4. The median of a triangle can be found also according to other data. For example, if lengths of two parties to one of which the median, for example, of length of the parties of AB and BC and also a corner x between them is carried out are set. Then length of a median can be found through the theorem of cosines: AE = sqrt ((AB^2+(BC^2)/4) - AB*BC*cos(x)).