Points in the Coordinate System

Describe points using coordinates and calculate distances between them

Overview

Points in a plane can be precisely described using coordinates. A coordinate system uses two perpendicular axes (the x-axis and y-axis) to define the position of any point.

Coordinate System Basics:

The coordinate system divides the plane into four quadrants. Points are described by an ordered pair of numbers (coordinates) that represent their distances from the axes.

Describing Points with Coordinates

Any point in a plane can be described by a pair of numbers, called coordinates.

Point Notation:

A point \(A\) is described as \(A(x, y)\), where:

  • \(x\) = horizontal distance from the y-axis (x-coordinate)
  • \(y\) = vertical distance from the x-axis (y-coordinate)

Sign Conventions

The signs of the coordinates depend on which quadrant the point is in:

  • Right of the y-axis: \(x\) is positive
  • Left of the y-axis: \(x\) is negative
  • Above the x-axis: \(y\) is positive
  • Below the x-axis: \(y\) is negative

Visual Example

Point coordinates in coordinate system

Distance Between Two Points

To find the distance between two points, we use the distance formula, which is derived from the Pythagorean theorem.

The Distance Formula

For two points \(A(x_1, y_1)\) and \(B(x_2, y_2)\), the distance between them is:

\(\displaystyle AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
Derivation:

This formula comes from the Pythagorean theorem. The distance between two points is the hypotenuse of a right triangle with legs equal to the differences in x and y coordinates.

Worked Example

Calculate Distance Between Two Points

Given: Point \(A(1, 2)\) and Point \(B(4, 5)\)

Find: The distance between points \(A\) and \(B\)

Step 1: Identify the coordinates

\(x_1 = 1, \quad y_1 = 2\)
\(x_2 = 4, \quad y_2 = 5\)

Step 2: Calculate the differences

\(\Delta x = x_2 - x_1 = 4 - 1 = 3\)
\(\Delta y = y_2 - y_1 = 5 - 2 = 3\)

Step 3: Apply the Pythagorean theorem

Distance calculation with right triangle

Step 4: Use the distance formula

\(AB^2 = AC^2 + BC^2 = 3^2 + 3^2 = 9 + 9 = 18\)

Step 5: Calculate the distance

\(\displaystyle AB = \sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2} \approx 4.243\)

Result: The distance between \(A\) and \(B\) is approximately \(4.243\) units (or exactly \(3\sqrt{2}\) units).

Key Concepts

  • Coordinates describe position: \(A(x, y)\)
  • Distance formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
  • Distance is always positive and symmetric: \(AB = BA\)
  • The distance formula is based on the Pythagorean theorem
  • Sign of coordinates indicates position relative to axes
  • Distance is measured in the same units as the coordinate axes

Learn More

For more detailed examples and exercises on distance calculations between points:

Distance Between Two Points Tutorial →


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