What is the distance formula for two points in Cartesian coordinates?

The distance formula for two points in Cartesian coordinates is: √((x₂ - x₁)² + (y₂ - y₁)²).

To understand this formula, let's break it down. Imagine you have two points on a graph, Point A (x₁, y₁) and Point B (x₂, y₂). The distance formula helps you find the straight-line distance between these two points. This formula is derived from the Pythagorean theorem, which is used in right-angled triangles.

First, you calculate the difference in the x-coordinates (x₂ - x₁) and the difference in the y-coordinates (y₂ - y₁). These differences represent the lengths of the two legs of a right-angled triangle. Squaring these differences (i.e., (x₂ - x₁)² and (y₂ - y₁)²) ensures that the values are positive and eliminates any negative signs.

Next, you add these squared differences together. This sum represents the square of the hypotenuse of the right-angled triangle formed by the points and the differences in their coordinates. Finally, taking the square root of this sum gives you the actual distance between the two points.

For example, if Point A is (3, 4) and Point B is (7, 1), the differences in coordinates are (7 - 3) = 4 and (1 - 4) = -3. Squaring these gives 16 and 9, respectively. Adding them together results in 25, and the square root of 25 is 5. So, the distance between the points (3, 4) and (7, 1) is 5 units.

This formula is a fundamental tool in geometry and is widely used in various applications, from simple graph plotting to complex engineering problems.