45. Geometry of position

Position Geometry, also known as Analytical Geometry, is one of the most important topics in the study of Mathematics for the Enem test. This branch of Mathematics, which combines aspects of geometry and algebra, is fundamental to understanding many complex mathematical concepts and problems.

Position Geometry is an extension of Euclidean geometry that uses numerical coordinates to represent and analyze the properties of geometric figures. It was developed by the French mathematician René Descartes in the 17th century, which is why it is sometimes called Cartesian Geometry, in his honor.

In essence, Position Geometry involves the representation of points, lines and figures on the Cartesian plane, which is a coordinate system that allows locating any point in space by means of two numbers, called coordinates. The Cartesian plane is divided into four quadrants by two perpendicular lines, called axes. The horizontal axis is called the abscissa axis (or x-axis), and the vertical axis is called the ordinate axis (or y-axis).

A point on the Cartesian plane is represented by a pair of numbers (x, y), where x is the coordinate of the point on the abscissa axis and y is the coordinate of the point on the ordinate axis. For example, the point (2, 3) represents the point that is two units to the right of the origin (the point where the axes intersect) and three units above the origin.

Lines in the Cartesian plane can be represented by linear equations, which are equations that express the relationship between the x and y coordinates of all the points that belong to the line. The most common form of a linear equation is y = mx + b, where m is the slope of the line and b is the y-intercept, which is the point where the line crosses the y-axis.

Position Geometry also lets you analyze and represent more complex geometric figures, such as circles, ellipses, parabolas, and hyperbolas. Each of these figures can be represented by a specific equation that expresses the relationship between the x and y coordinates of all the points that belong to the figure.

For example, the equation of a circle with center at the origin and radius r is x² + y² = r². This equation expresses the fact that the distance from any point (x, y) on the circle to the origin is equal to the radius of the circle.

Another important concept in Position Geometry is the distance between two points. The distance formula, which is derived from the Pythagorean theorem, allows you to calculate the distance between two points (x1, y1) and (x2, y2) in the Cartesian plane: d = √[(x2 - x1)² + (y2 - y1 )²].

In summary, Position Geometry is a powerful tool for the representation and analysis of geometric figures. Your understanding is crucial for solving many Mathematics problems on the Enem, especially those involving geometry and algebra.

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