Distance Formula

What is the Distance Formula?

The distance formula allows us to calculate the distance between two points in a Cartesian plane. These points, represented as $(x_1, y_1)$ and $(x_2, y_2)$, have a straight-line distance given by:

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

How is it Derived?

This formula is derived from the Pythagorean theorem. Imagine a right triangle formed by the line connecting $(x_1, y_1)$ and $(x_2, y_2)$, with horizontal and vertical legs of lengths $|x_2 - x_1|$ and $|y_2 - y_1|$. The hypotenuse of this triangle represents the distance $d$.

Applying the Pythagorean theorem:

$$d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2$$

Taking the square root gives:

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

When to Use the Distance Formula?

Use the formula whenever you need to find the length of a straight line between two points in the Cartesian plane. This is useful in:

• Geometry problems.
• Graphical analysis.
• Applications like physics, navigation, and more.

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Worked Example

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Summary

• The distance formula calculates the straight-line distance between two points:

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

• Derived from the Pythagorean theorem.
• Steps to apply the formula:
  1. Identify the coordinates of the points, $(x_1, y_1)$ and $(x_2, y_2)$
  2. Substitute values into the formula.
  3. Simplify to find $d$
• Practice using the formula with real-world and mathematical problems.