To calculate the distance between two points in a Cartesian coordinate system, specifically the points (a, b) and (c, d), we can use the distance formula derived from the Pythagorean theorem. The formula is:
Distance = √[(c – a)² + (d – b)²]
Here’s a more detailed breakdown of the formula:
- (c – a): This part of the formula represents the difference in the x-coordinates of the two points. It measures how far apart the points are along the x-axis.
- (d – b): This measures the difference in the y-coordinates of the two points, reflecting the vertical distance between them on the y-axis.
- (c – a)² + (d – b)²: By squaring these differences, we ensure that we always get a positive value, regardless of whether the points are in the first, second, third, or fourth quadrant of the Cartesian plane.
- √: Finally, by taking the square root of the sum, we find the actual distance between the two points in a straight line, which is often referred to as the Euclidean distance.
For example, if we want to calculate the distance between the points (1, 2) and (4, 6), we would plug them into the formula as follows:
Distance = √[(4 - 1)² + (6 - 2)²]
Distance = √[3² + 4²]
Distance = √[9 + 16]
Distance = √25
Distance = 5Therefore, the distance between the points (1, 2) and (4, 6) is 5 units.
Using this formula is essential in various fields, including mathematics, physics, engineering, and computer science, as it provides a fundamental way to measure distances between two points in a two-dimensional space.