The median in a triangle is a piece which is carried out from vertex of angle to the middle of the opposite side. To find median length, it is necessary to use a formula of its expression through all parties of a triangle which is easy for removing.
Instruction
1. To remove a formula for a median in any triangle, it is necessary to address the investigation from the theorem of cosines for the parallelogram which is turning out by completion of a triangle. The formula can be proved on this basis, it is very convenient at the solution of tasks if all lengths of the parties are known or them it is easily possible to find from other initial data of a task.
2. Actually the theorem of cosines represents generalization of Pythagorean theorem. It sounds so: for a two-dimensional triangle with lengths of the parties of a, b and c and a corner α, opposite to the party a, fairly following equality: a² = b² + with² – 2•b • with • cos α.
3. The generalizing investigation from the theorem of cosines defines one of the most important properties of a quadrangle: the sum of squares of diagonals is equal to the sum of squares of all its parties: d1² + d2² = a² + b² + with² + d².
4. Solve a problem: let in any triangle of ABC all parties be known, find its median BM.
5. Complete a triangle to ABCD parallelogram addition of the lines parallel to an and c. thus, the figure with the parties of an and c and diagonal of b was created. It is the most convenient to build so: postpone on continuation of a straight line which possesses a median, a piece of the MD same length, connect its top to tops of the remained two parties of A and C.
6. On property of a parallelogram of diagonal are divided a point of intersection into equal parts. Apply the investigation from the theorem of cosines according to which the sum of squares of diagonals of a parallelogram is equal to the sum of the doubled squares of its parties: BK² + AC² = 2•AB² + 2•BC².
7. As BK = 2•BM, and BM is a median of m: (2 · m)² + b² = 2 • with² + 2•a², from where: m = 1/2 · √ (2 • with² + 2•a² - b²).
8. You removed a formula of one of triangle medians for the party of b: mb = m. Similarly there are medians of two other its parties: ma = 1/2 • √ (2 • with² + 2•b² - a²); mc = 1/2 • √ (2•a² + 2•b² - with²).