Figure perimeter – the sum of lengths of all its parties. Respectively, to find triangle perimeter, it is necessary to know what length of each of its parties is equal to. For search of the parties properties of a triangle and the main theorems of geometry are used.
Instruction
1. If all three parties of a triangle are already given in a statement of the problem, just put them. Then the perimeter will be equal: P = a + b + with.
2. Let two parties of a, b and a corner between them γ be given. Then the third party can be found according to the theorem of cosines: with² = a² + b² - 2 • · b • cos(γ). Remember that length of the party can be only positive.
3. A special case of the theorem of cosines – Pythagorean theorem which is applicable for rectangular triangles. The corner γ in this case is equal 90 °. The cosine of a right angle addresses in unit. Then with² = a² + b².
4. If in a condition only one of the parties is given, but at the same time triangle corners are known, two other parties can be found according to the theorem of sine. By the way, corners can be set not everything therefore it is useful to remember that the sum of all corners of a triangle is equal 180 °.
5. So, let the party of a, a corner γ between an and b, β between an and c is given. α between the parties of b and c it is easy to find the third corner from the theorem of the sum of corners of a triangle: α = 180 ° - β – γ. According to the theorem of sine, a/sin(α) = b/sin(β) = with / sin(γ) = 2 • R where R is the radius of a circle, circumscribed about a triangle. To find the party of b, it is possible to express it from this equality through corners and the party of a: b = a • sin(β)/sin(α). Similarly also the party of c is expressed: c = a • sin(γ)/sin(α). If, for example, the radius of a circumscribed circle is given, but length of any of the parties is not given, it is also possible to solve a problem.
6. If in a task the area of a figure is given, it is necessary to write down a formula for the area of a triangle through the parties. The choice of a formula depends on what else is known. If, besides the area, two parties are set, application of a formula of Heron will help. The area can be expressed also through two parties and a sine of the angle between them: S = 1/2 • · b • sin(γ), where γ – a corner between the parties of an and b.
7. In some tasks the area and radius of a circle, inscribed in a triangle can be set. In that case formula r = will help out S/p where r is the radius of an inscribed circle, S is the area, p – poluperimetr a triangle. Poluperimetr from this formula it is easy to express: p = S / r. It was necessary to find perimeter: P = 2 • p.