Analytical Geometry: Distance between Points

Relevance of the Topic

The distance between two points in a plane or in space is a critical concept in Analytical Geometry and has extensive applications in both pure mathematics and in science and engineering. It is a fundamental ingredient in many areas, such as vector calculus, space geometry, physics, computer science, and mathematical modeling. Understanding the distance between two points is essential for grasping the structure of geometric spaces and for solving problems involving physical and metric quantities.

Contextualization

The distance between two points is one of the first notions introduced in Geometry. It is part of the initial study of Analytical Geometry, which is an area of mathematics that combines algebra with geometry. This study is particularly relevant in the 3rd year of High School, as it prepares students for more advanced topics in mathematics, including calculus. Furthermore, the distance between two points is an interdisciplinary topic, extending to various areas such as physics, engineering, and computer science. Therefore, a complete and practical understanding of this concept is a necessary propaedeutic for the development of mathematical reasoning and logical thinking.

Theoretical Development

• Distance between two points in the Cartesian plane:

  • Given two points in the Cartesian plane, A(x1, y1) and B(x2, y2), the formula to calculate the distance between them is:
    • d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
  • In other words, the distance between two points is the square root of the sum of the squares of the differences of their x and y coordinates.

• Explanation of the formula:

  • To understand the formula, visualize a triangle formed by points A, B, and point C, which is the vertex opposite to segment AB. The coordinates of C are (x2, y1).
  • Now, consider the base of the triangle, segment AC. Its length is given by the difference between the x coordinates: (x2 - x1).
  • The height of the triangle, that is, the distance from point B to segment AC, is equal to the difference of the y coordinates: (y2 - y1).
  • Therefore, the distance formula is actually a generalization of the Pythagorean theorem, applied to this right triangle.
  • The square root is used to return the distance as a positive number.

• Application Examples:

  • Calculate the distance between points A(3, 4) and B(8, 9):
    • d = sqrt((8 - 3)^2 + (9 - 4)^2) = sqrt(25 + 25) = sqrt(50) ~ 7.07
    • The distance is approximately 7.07 units.
  • Determine if points C(-2, 1) and D(6, -3) are at the same distance from A and B:
    • The distance between A and C is sqrt((3 - (-2))^2 + (4 - 1)^2) = sqrt(25) = 5.
    • The distance between A and D is sqrt((3 - 6)^2 + (4 - (-3))^2) = sqrt(74).
    • Therefore, C is at the same distance from A and B, while D is at a different distance.

Detailed Summary

• Key Points:

  • Concept of coordinates: The location of a point in the plane is given by an ordered pair of real numbers, (x, y), called the coordinates of the point.
  • Utility of the Pythagorean theorem: It is often used in calculating the distance between two points, as the distance is the length of the hypotenuse of a right triangle.
  • Calculation dynamics: The distance between two points is obtained by calculating the difference of the x and y coordinates, squaring each difference, summing the results, and finally taking the square root of this sum.

• Key Terms:

  • Coordinates: Numbers that describe the position of a point in space. In the Cartesian plane, coordinates are ordered pairs, (x, y).
  • Distance: The amount of space between two points. In the Cartesian plane, the formula to calculate the distance between two points is based on the Pythagorean theorem.

• Conclusions:

  • The distance between two points in a plane or in space can be calculated using the distance formula, which is a generalization of the Pythagorean theorem.
  • The distance formula is crucial in Analytical Geometry and has extensive applications in various areas of mathematics and applied sciences.
  • Detecting if four points are at the same distance can be done by calculating the distance between each pair of points and comparing the results.

• Exercises:

  1. Calculate the distance between points A(5, 7) and B(-3, 1).
  2. Verify if points C(1, 3) and D(-5, -1) are at the same distance from A and B.
  3. Determine if points E(0, 0) and F(8, 6) are at the same distance from A and B.