Distance between Two Points Formula

The distance between two points on a coordinate plane is the straight-line distance connecting them, calculated using their \( x \) and \( y \) coordinates. This distance formula is derived from the Pythagorean Theorem and is useful in geometry, physics, and various applications where distances need to be measured on a plane.

Formula for the Distance between Two Points

If we have two points \( (x_1, y_1) \) and \( (x_2, y_2) \) on a coordinate plane, the distance \( d \) between these points can be calculated with the formula:

\( d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2} \)

In this formula:

  • \( x_1 \) and \( y_1 \) are the \( x \)- and \( y \)-coordinates of the first point
  • \( x_2 \) and \( y_2 \) are the \( x \)- and \( y \)-coordinates of the second point
  • \( d \) is the distance between the two points

Detailed Explanation of the Formula

The formula \( d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2} \) is based on the Pythagorean Theorem. When calculating the distance between two points, we consider the horizontal and vertical differences as the two legs of a right triangle, and the distance \( d \) as the hypotenuse.

  • First, calculate the difference in the \( x \)-coordinates: \( x_2 – x_1 \).
  • Next, calculate the difference in the \( y \)-coordinates: \( y_2 – y_1 \).
  • Square both differences and add them together.
  • Finally, take the square root of the sum to find the distance \( d \).

Example 1: Calculating Distance between Two Points

Problem: Find the distance between the points \( (2, 3) \) and \( (7, 8) \) on a coordinate plane.

Solution:

  1. Write down the formula: \( d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2} \).
  2. Substitute \( x_1 = 2 \), \( y_1 = 3 \), \( x_2 = 7 \), and \( y_2 = 8 \): \( d = \sqrt{(7 – 2)^2 + (8 – 3)^2} \).
  3. Calculate the differences: \( d = \sqrt{5^2 + 5^2} \).
  4. Square the differences: \( d = \sqrt{25 + 25} \).
  5. Add and take the square root: \( d = \sqrt{50} \approx 7.07 \).

The distance between the points \( (2, 3) \) and \( (7, 8) \) is approximately \( 7.07 \) units.

Example 2: Using the Distance Formula in Real-Life Context

Problem: A city park is represented on a map with coordinates for two playgrounds: one at \( (4, 1) \) and the other at \( (-3, 5) \). Find the distance between these two playgrounds.

Solution:

  1. Write down the formula: \( d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2} \).
  2. Substitute \( x_1 = 4 \), \( y_1 = 1 \), \( x_2 = -3 \), and \( y_2 = 5 \): \( d = \sqrt{(-3 – 4)^2 + (5 – 1)^2} \).
  3. Calculate the differences: \( d = \sqrt{(-7)^2 + 4^2} \).
  4. Square the differences: \( d = \sqrt{49 + 16} \).
  5. Add and take the square root: \( d = \sqrt{65} \approx 8.06 \).

The distance between the two playgrounds is approximately \( 8.06 \) units.

This formula is widely used in geometry and other fields to find the straight-line distance between two points on a plane, using their coordinates.