analytic geometry is the branch of Mathematics that studies the plane geometry and space through algebraic processes. This means that the whole geometryEuclidean can be studied through the procedures established by the geometryanalytical. In this way, she creates for Euclidean geometry new techniques that can be used for proving theorems, creating and proving properties, etc.
The foundations of analytic geometry
The first step to be taken to study the geometryEuclidean (flat and spatial), through Law Suitalgebraic, is to create mechanisms to introduce the algebra in that discipline. For this purpose, the number line is used so that specific points represent real numbers unique. So, the distance between any point of the number line and its origin is a real number relative to the location of that point on the line. This real number can be called the point coordinate.
taking two straights perpendicular that are at the origin, it is possible to find the location of any point within the plane formed by them using an ordered pair, which is the set of two coordinates, each relative to one of the lines they defined that
flat. The same goes for three orthogonal lines that meet at their origins: they form a three-dimensional space, in which it is possible to determine the location of any point by means of ordered terms.O flat described above, formed by two perpendicular lines that meet at their origins, is called flatCartesian. This plan is the first space in which we study the geometryanalytical.
so much in straight how much in the flat and in the space, it is possible to define the distance between two points. That distance is defined as the length of the straight segment that connects them. Now imagine a Cartesian plane and on it the points A(0, 0), B(0, 1), C(1, 1) and D(1, 0). These points form a square, and this can be seen in the following figure:
The internal angles of the figure formed by the points above are all straight, and the distance between two consecutive points is always equal to 1 unit.
Therefore, the concept of distancein betweentwopoints is one of the most important of the entire geometryanalytical. This concept allows from the definition of some elements, such as the length of the line segment, to the demonstration of important theorems of Geometry.
Distance between two points
As stated before, the concept of distancein betweentwopoints is one of the most important of geometryanalytical. In the square in the previous image, the distances shown were straight lines parallel to the x-axis or y-axis, but it is possible to calculate the distance between any two points on a Cartesian plane.
For that, let's turn to algebra. Given the points A(xTHEyTHE) and B(xByB), we know that the distance between these two points is the length of segment AB. Note this segment in the following figure:
The projections of points A and B onto the axes form triangle ABC, which is a rectangle in C. Note that the length of segment AC is equal to xB – xTHE, and that the length of segment BC is given by yB - yTHE. The length of segment AB can be obtained by means of Pythagorean theorem:
This result obtained is the formula for calculating the distancein betweentwopoints on the plan.
By Luiz Paulo Moreira
Graduated in Mathematics
Source: Brazil School - https://brasilescola.uol.com.br/o-que-e/matematica/o-que-e-geometria-analitica.htm