The way of calculation of the unknown party of a triangle depends not only on task conditions, but also on for what it becomes. Not only school students at geometry lessons, but also the engineers working in the different industries of production, interior designers, cutters and representatives of many other professions face a similar task. Accuracy of calculations for the different purposes can be different, but their principle remains to the same, as in the school book of problems.
It is required to you
- - a triangle with the set parameters;
- - calculator;
- - handle;
- - pencil;
- - protractor;
- - sheet of paper;
- - computer with the AutoCAD program;
- - theorems of sine and cosines.
1. Draw the triangle corresponding to task conditions. The triangle can be constructed on three parties, two parties and a corner between them or to the party and two corners adjoining to it. The principle of work in a notebook and on the computer in the AutoCAD program in this plan are identical. So in a task the sizes of one or two parties and one or two corners have to be specified.
2. At construction on two parties and a corner draw the piece equal to the known party on a leaf. By means of a protractor postpone the set corner and carry out the second party, having postponed the size given in a condition. If you were given one party and two corners, adjacent to it, draw at first the party, then from two ends of the received piece postpone corners and carry out two other parties. Designate a triangle as ABC.
3. In the AutoCAD program it is the most convenient to build the wrong triangle by means of the Piece tool. You will find it through main tab, having chosen the Drawing window. Set coordinates of the party known to you, then — a final point of the second set piece.
4. Define a type of a triangle. If it rectangular, then the unknown party is calculated on Pythagorean theorem. The hypotenuse is equal to a square root from the sum of squares of legs, that is c= √ to a2+b2. Respectively, any of their legs will be to equally square root from the difference of squares of a hypotenuse and the known leg: a= √ c2-b2.
5. For calculation of the unknown party of a triangle at which the party and two adjacent corners are given use the theorem of sine. The party and so treats sinα as the party of b to sinβ. Α and β in this case — opposite corners. The corner which is not set by statements of the problem can be found, having remembered that the sum of internal corners of a triangle is equal 180 °. Subtract from it the sum of two corners known to you. Find the party of b unknown to you, having solved a proportion in the usual way, that is having increased the known party and by sinβ and having divided this work into sinα. You receive a formula b=a*sinβ/sinα.
6. If the parties of an and b and a corner γ between them are known to you, use the theorem of cosines. The unknown party with will be equal to a square root from the sum of squares of two other parties, minus the doubled work of the same parties increased by a cosine of the angle between them. That is c= √ a2+b2-2ab*cosγ.