What is the formula for the distance between two points?

The formula for the distance between two points is the square root of the sum of the squares of the differences in their coordinates.

To find the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) on a coordinate plane, you use the distance formula. This formula is derived from the Pythagorean theorem, which relates the sides of a right-angled triangle. Imagine a right-angled triangle where the horizontal leg is the difference in the x-coordinates \((x_2 - x_1)\) and the vertical leg is the difference in the y-coordinates \((y_2 - y_1)\). The distance between the two points is the hypotenuse of this triangle.

The distance formula is written as:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Here’s how it works step-by-step:
1. Subtract the x-coordinate of the first point from the x-coordinate of the second point to find the horizontal distance: \( x_2 - x_1 \).
2. Subtract the y-coordinate of the first point from the y-coordinate of the second point to find the vertical distance: \( y_2 - y_1 \).
3. Square both of these differences: \((x_2 - x_1)^2\) and \((y_2 - y_1)^2\).
4. Add these squared differences together: \((x_2 - x_1)^2 + (y_2 - y_1)^2\).
5. Finally, take the square root of this sum to find the distance: \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).

This formula gives you the straight-line distance between the two points, which is also known as the Euclidean distance. It’s a fundamental concept in geometry and is widely used in various fields such as physics, engineering, and computer science.