Irrespective of its look, it is possible to enter only one circle in each triangle. Its center at the same time is also a point of intersection of bisectors. The rectangular triangle has a number of own properties which need to be considered at calculation of radius of an inscribed circle. Data in a task can be specified different and there is a need to carry out additional calculations.
It is required to you
- - a rectangular triangle with the set parameters;
- - pencil;
- - sheet of paper;
- - ruler;
- - compasses.
Instruction
1. Begin with construction. Draw a triangle with given sizes. Any triangle is under construction on three parties, the party and two corners or two parties and a corner between them. As the size of one corner is set initially, in conditions either two legs, or one of legs and one of corners, or one leg and a hypotenuse have to be specified. Designate a triangle as DIA where With — top of a right angle. Designate opposite to corners legs as and and b, and a hypotenuse — as page. Designate the radius of an inscribed circle as r.
2. To have an opportunity to apply a classical formula of calculation of radius of an inscribed circle, find all three parties. The way of calculations depends on what is set in conditions. If the sizes of all three parties are given, calculate poluperimetr on formula p=(a+b+c)/2. If you were given the sizes of two legs, find a hypotenuse. According to Pythagorean theorem, it is equal to a square root from the sum of squares of legs, that is with = √ to a2+b2.
3. When one leg and a corner is given, define whether it is opposite or adjacent. In the first case use the theorem of sine, that is find a hypotenuse on a formula with =a/sinCAB, in the second — consider according to the theorem of cosines. In this case with =a/cosCBA. Having executed calculations, find poluperimetr a triangle.
4. Knowing poluperimetr, it is possible to calculate the radius of an inscribed circle. It is equal to a square root from fraction in which numerator there is a poizvedeniye of differences of this poluperimetr with all parties, and in denominators — poluperimetr. That is r= √ (p-a) (p-b) (p-c)/p.
5. Pay attention that the numerator of this radicand represents the area of this triangle. That is radius can be found and some other way, having divided the area on poluperimetr. So if both legs are known, then calculations become simpler a little. It is necessary for a poluperimetr a hypotenuse find on the sum of squares of legs. Count the area, having increased legs at each other and having divided the received number into 2.