To find the length of the base of the triangle at the sides without any additional parameters only if they are represented by their coordinates in a two - or three-dimensional system. For example, suppose the three-dimensional coordinates of the point A(X₁,Y₁,Z₁), B(X₂,Y₂,Z₂) and C(X₃,Y₃,Z₃), the segments between them form the side. Then, you know, and the coordinates of the third side (the base) is formed by the segment AC. To calculate its length, find the difference between the coordinates of the points along each axis, the values obtained erect in the square, and fold, and the result extract the square root of a: AC = √((X₃-X₁)2 + (Y₃-Y₁)2 + (Z₃ Is Z₁)2).
If you know only the length of each of the sides (a) to calculate the length of the base (b) need more information - for example, the magnitude of the angle between them (γ). In this case you can use the theorem of cosines, which implies that the length of a side of a triangle (not necessarily isosceles) is equal to the square root of the sum of the squares of the lengths of two other sides, from which is subtracted twice the product of their lengths into the cosine of the angle between them. Since in an equilateral triangle the length of a formula involved parties are the same, it can be simplified: b = a*√(2*(1-cos(γ))).
For the same initial data (the length of the sides equal to a, the angle between them is equal to γ) can be used and the theorem of sines. To do this, find twice the product of the known length of side to the sine of half the angle lying opposite the base of the triangle: b = 2*a*sin(γ/2).
If in addition the lengths of the sides (a) given the angle (α) adjacent to the base, we can apply the theorem on projections: the length of a side is equal to the sum of the other two sides into the cosine of the angle which they form with the party. Since in an equilateral triangle these sides, as the angles involved are of the same magnitude, then write the formula: b = 2*a*cos(α).
Advice 2: How to find the length of a side of a triangle by coordinates
Geometric tasks of any level of high level of complexity require the presence of a person's ability to solve elementary problems. Otherwise, the possibility of obtaining the desired result is greatly reduced. In addition to the process is almost intuitive feeling of the correct way, leading to the desired result, you need to be able to calculate area, know a large number of auxiliary theorems, freely to carry out calculations in the coordinate plane.
Use the formula to calculate the cut length, if your task explicitly set the coordinates of the vertices of the triangle. For this follow some simple steps. First calculate the difference between the coordinates of corresponding points on the x-axis and the y-axis. The results obtained bring into a square and summarize. The square root of the resulting value is the required cut length.
Consider all these tasks, if no data to solve simple tasks. Separately write down everything listed in the condition. Pay attention to the type of the described triangle. If it is rectangular, then you need to know the coordinates of two vertices: the length of the third side, you will be able to find, using the formula of Pythagoras. Also simplifies the situation when working with isosceles or equilateral triangle, mi.
Pay attention to some of the characteristic elements conditions, which contain a clue. For example, the text may be mentioned that the vertex of the triangle lies on one of the axes (which gives you information about one of the coordinate), passes through the origin. All it's important to write to have full information.
Don't forget about the formulas used to Express the sides of a triangle through the other elements, and existing proportional relationship. The minimum number of auxiliary equations that will be useful include formulas for finding the height, medians, and bisectors of triangles. In addition, remember that the two sides of a triangle are in the same relation to each other as the segments into which divides the bisector drawn to the third side.
Be prepared for the fact that if you use in the solution of certain formulas or theorems you may be asked to prove them or to describe the procedure output.