The party of a triangle is the straight line limited to its tops. In total them at a figure three, this number defines amount practically of all graphic characteristics: corner, median, bisector, etc. To find the party of a triangle, it is necessary to study attentively entry conditions of a task and to define what of them can become the main or intermediate sizes for calculation.
Instruction
1. The parties of a triangle, as well as other polygons, have own names: sides, the basis and also a hypotenuse and legs at a figure with a right angle. It facilitates calculations and formulas, making their more clear even if a triangle any. A graphic figure therefore it it is always possible to arrange so that to make the solution of a task more evident.
2. The parties of any triangle are connected among themselves and other its characteristics by various ratios which help to calculate the required size in one or several actions. At the same time the more difficult the task, the is longer the sequence of steps.
3. The decision becomes simpler if a triangle standard: the words "rectangular", "isosceles", "equilateral" mark out a certain interrelation between its parties and corners at once.
4. Lengths of the parties in a rectangular triangle are connected among themselves by Pythagorean theorem: the sum of squares of legs is equal to a hypotenuse square. And corners, in turn, are connected with the parties the theorem of sine. It approves equality of the relations between lengths of the parties and the trigonometrical sin function of an opposite corner. However, it is right for any triangle.
5. Two parties of an isosceles triangle are equal among themselves. If their length is known, one more size suffices to find a third. For example, let height which is carried out to it is known. This piece divides the third party into two equal parts and allocates two rectangular triangles. Having considered one of them, on Pythagorean theorem find a leg and increase by 2. It will also be length of the unknown party.
6. The party of a triangle can be found through other parties, corners, lengths of height, a median, bisector, size of perimeter, the area, radius of an inscribed circle, etc. If it is impossible to apply one formula at once, then make a number of intermediate calculations.
7. Review an example: find the party of any triangle, knowing ma=5 median which is carried out to it and lengths of two other medians of mb=7 and mc=8.
8. ReshenieZadach is assumed by use of formulas for a median. It is necessary to find the party and. Obviously, it is necessary to work out three equations with three unknown.
9. Write down formulas for all medians: ma = 1/2 • √ (2 • (b² + with²) – a²) = 5; mb = 1/2 • √ (2 • (a² + with²) – b²) = 7; mc = 1/2 • √ (2 • (a² + b²) – with²) = 8.
10. Express with² from the third equation and set up her in the second: with² = 256 - 2 ·a² – 2•² b² = 20 → with² = 216 – a².
11. Square both parties of the first equation and find a, having entered the expressed sizes: 25 = 1/4 · (2•20 + 2 • (216 – a²) – a²) → a ≈ 11.1.