The triangle is made by three pieces connected by the extreme points. Finding of length of one of these pieces - the parties of a triangle - very widespread task. Knowledge of only lengths of two parties of a figure is not enough for calculation of length of the third, one more parameter is for this purpose necessary. It can be corner size in one of figure tops, its area, perimeter, radius of the entered or described circles, etc.
Instruction
1. If it is known that the triangle is rectangular, it gives you knowledge of size of one of corners, i.e. lacking for calculations of the third parameter. The required party (C) can be a hypotenuse - the party lying opposite to a right angle. Then for its calculation take a square root and the squared and put lengths of two other parties (A and B) this figure: C= √ (A²+B²). If the required party is a leg, take a square root from the difference between squares of lengths of bigger (hypotenuse) and smaller (the second leg) the parties: C= √ (A²-B²). These formulas follow from Pythagorean theorem.
2. Knowledge as the third parameter of perimeter of a triangle (P) reduces a problem of calculation of length of the missing party (C) to the simplest operation of subtraction - take away from perimeter of length both (A and B) the known parties of a figure: C=P-A-B. This formula follows from determination of perimeter which is length of the broken line limiting the area of a figure.
3. Existence in initial conditions of size of a corner (γ) between the parties (A and B) the known length will demand for finding of length of the third (C) calculation of trigonometrical function. Square both lengths of the parties and put results. Then subtract the work of their lengths on a cosine of the known corner from the received value, and in conclusion take a square root from the received size: With = √ (A²+B²-A*B*cos(γ)). The theorem which you used in calculations is called the theorem of sine.
4. Izvestnaya Square of a triangle (S) will demand use determines the area as a half of the work of length of the known parties (A and B) on a sine of the angle between them. Express from it a sine of the angle, and you receive expression 2*S / (A*B). The second formula will allow to express a cosine of the same corner: as the sum of squares of a sine and cosine of an identical corner is equal to unit, the cosine is equal to a root from a difference between unit and a square of the expression received earlier: √ (1-(2*S/(A*B))²). The third formula - the theorem of cosines - was used in the previous step, replace in it a cosine with the received expression and you will have such formula for calculation: With = √ (A²+B²-A*B * √ (1-(2*S/(A*B))²)).