Advice 1: How to find angle of triangle given by its coordinates

If you know the coordinates of all three vertices of the triangleof a square, you can find its angles. The coordinates of a point in three-dimensional space - x,y, and z. However, three points which are vertices of the triangleof the polygon, we can always draw a plane, so in this problem it is more convenient to consider only two coordinates, x and y, considering the z coordinate for all points is the same.
You will need
• The coordinates of the triangle
Instruction
1
Let the point A of the trianglepolygon ABC has coordinates x1, y1, point B of this tregon - coordinates x2, y2 and point C coordinates x3, y3. What are the x and y coordinates of the vertices of the triangleof a square. In a Cartesian coordinate system with mutually perpendicular axes X and Y from the origin it is possible to hold the radius-vectors for all three points. The projection of the radius-vectors on the coordinate axes and will give the coordinates of the points.
2
Let r1 is the radius vector of point A, r2 is the radius-vector of point B, and r3 is the radius-vector of the point C.
It is obvious that the length of the side AB is equal to |r1-r2|, the length of side AC = |r1-r3|, a, BC = |r2-r3|.
Therefore AB = sqrt(((x1-x2)^2)+((y1-y2)^2)), AC = sqrt(((x1-x3)^2)+((y1-y3)^2)), BC = sqrt(((x2-x3)^2)+((y2-y3)^2)).
3
The angles of the trianglepolygon ABC can be found from the spherical law of cosines. The theorem of cosines can be written in the following form: BC^2 = (AB^2)+(AC^2) - 2AB*AC*cos(BAC). Hence, cos(BAC) = ((AB^2)+(AC^2)-(BC^2))/2*AB*AC. After substituting in this expression the coordinates, you get: cos(BAC) = (((x1-x2)^2)+((y1-y2)^2)+((x1-x3)^2)+((y1-y3)^2)-((x2-x3)^2)-((y2-y3)^2))/(2*sqrt(((x1-x2)^2)+((y1-y2)^2))*sqrt(((x1-x3)^2)+((y1-y3)^2)))

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.
Instruction
1
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.
2
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.
3
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.
4
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.
5
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.
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