Summary of Distance Between Points in the Cartesian Plane

Introduction

Relevance of the Theme

The distance between points on the Cartesian plane is a fundamental milestone in Mathematics, being one of the first steps to explore Analytical Geometry. Its understanding is vital for subsequent topics, such as the equation of a line, the proof of triangle congruence, and calculations of areas and perimeters.

Contextualization

Within the vast universe of Mathematics, Analytical Geometry presents itself as a valuable tool for the description and analysis of shapes and figures in space. The distance between points, in this context, allows for an exact and absolute measure between any two elements of the plane, regardless of their location. Therefore, this topic is a bridge that connects algebra (Cartesian coordinates) and geometry, making a smooth transition between numerical and visual spaces.

Theoretical Development

Components

• Cartesian Plane: An essential tool for graphically representing points in space. It is composed of two perpendicular axes - the x-axis (abscissas) and the y-axis (ordinates) - which intersect at the origin point (0,0).

• Coordinates: The location of a point on the Cartesian plane is given by an ordered pair, (x, y), where the first value refers to the position on the x-axis and the second value, to the y-axis. The coordinates uniquely define the location of the point on the plane.

• Distance between Two Points: Having the ordered pairs indicating the points' locations at hand, the distance between them can be calculated. This is the main focus of our study.

Key Terms

• Distance: The length of the straight line that connects two points. It can be calculated using the Pythagorean theorem in a triangle with sides specified by the points' coordinates.

• Pythagorean Theorem: The theorem that establishes a fundamental relationship between the sides of a right triangle, expressed as (hypotenuse)^2 = (leg)^2 + (leg)^2.

• Triangle Legs: In the context of distance calculation, the legs are the line segments that connect one of the points in the plane to the projection of the other on the corresponding axis. It is the lengths of the legs that we apply in the Pythagorean theorem to find the distance.

Examples and Cases

• Example 1: Let A(3,1) and B(7,4) be the points. To calculate the distance between them, we draw the legs, which are line segments perpendicular to the sides of the formed triangle. The legs have lengths of 4 (3-7) and 3 (1-4), respectively. Applying the Pythagorean theorem: (4)^2 + (3)^2 = (distance)^2. The result is (distance)^2 = 25, therefore the distance between A and B is 5 units.

• Example 2: Assuming points C(-2,3) and D(1,-1), we again apply the Pythagorean theorem to find the distance. The legs have lengths of 3 (-2-1) and 4 (3-(-1)), respectively. (3)^2 + (4)^2 = (distance)^2. The result is (distance)^2 = 25, so the distance between C and D is also 5 units.

These examples illustrate the versatility of distance calculation, regardless of the location and orientation of the points on the plane, it will always be possible to find a precise measure of the separation between them.

Detailed Summary

Key Points

• Definition of Points on the Cartesian Plane: A point is defined by an ordered pair of real numbers, (x, y), where the first number refers to the position on the x-axis and the second number, to the y-axis. Each point on the plane is unique and can be identified by its coordinates.

• Use of Ordered Pairs for Distance Calculation: The distance between two points on a Cartesian plane can be calculated using the ordered pairs indicating the coordinates of each point.

• Pythagorean Theorem for Distance Calculation: The Pythagorean theorem, originating from Euclidean Geometry, is a key tool for calculating distances on a Cartesian plane. The theorem establishes that in a right triangle, the sum of the squares of the legs (the sides adjacent to the right angle) is equal to the square of the hypotenuse (the side opposite the right angle).

• Importance of Distance: The concept of distance is fundamental in many areas of Mathematics, especially in Geometry. It is the metric used to measure the space between two points or two objects.

Conclusions

• Precise Calculation of Distance: The distance between any two points on a Cartesian plane can be calculated precisely and directly using the Pythagorean theorem.

• Universality of Distance Calculation: Regardless of the location or orientation of the points on the plane, distance calculation is always possible, demonstrating the applicability and versatility of this concept.

• Points as Isolated Elements with Connections: Although points are represented as isolated elements on the Cartesian plane, the distance between them establishes a connection, a metric that unites them.

Suggested Exercises

1. Exercise 1: Given points E(2,4) and F(-3,1), calculate the distance between them using the Pythagorean theorem.

2. Exercise 2: Find the distance between points G(0,0) and H(5,-3), using the same strategy as the previous exercise.

3. Exercise 3: If points I(1,2) and J(4,5) represent the endpoints of a line segment, can we affirm that this segment has the same distance between its endpoints as the distance between points E(2,4) and F(-3,1)? Justify your answer.