Understanding the Formula for the Distance Between Two Points

The distance between two points in a two-dimensional plane can be calculated using a straightforward mathematical formula. This formula, known as the distance formula, has profound geometric implications and is derived from the principles of coordinate geometry and the Pythagorean theorem. In this article, we will explore the concept, the formula, and the derivation of the distance formula.

Formula for Distance Between Two Points

The formula for the distance between two points x1, y1 and x2, y2 in a two-dimensional Cartesian coordinate system is given by:

d sqrt[(x2 - x1)2 (y2 - y1)2]

Here, d represents the distance between the two points.

Three Dimensions

In three-dimensional space, the formula extends to include the z-coordinates. The distance formula for three-dimensional points x1, y1, z1 and x2, y2, z2 is:

d sqrt[(x2 - x1)2 (y2 - y1)2 (z2 - z1)2]

Derivation of the Distance Formula

The distance formula is derived from the Pythagorean theorem. Let's consider two points A(x1, y1) and B(x2, y2) in a two-dimensional coordinate plane as illustrated in the diagram below:

Diagram of points A and B with a triangle formed by connecting them to a common point C

Connect points A and B directly, forming a slanting line AB. Now, complete the triangle by joining points A and B to a common point C such that AC is parallel to the y-axis and BC is parallel to the x-axis.

The coordinates of point C can be derived as follows:

Since AC is parallel to the y-axis, the x-coordinate of C is the same as the x-coordinate of A, i.e., x1 Since BC is parallel to the x-axis, the y-coordinate of C is the same as the y-coordinate of B, i.e., y2

The coordinates of point C are therefore C(x1, y2).

On the x-y plane, the x-axis and y-axis intersect at a right angle. Therefore, AC and BC form a right-angled triangle ACB, with AB as the hypotenuse, AC as the perpendicular, and BC as the base.

Using the Pythagorean theorem:

AB2 AC2 BC2

From the diagram:

|y2 - y1| (y2 - y1)2 BC |x2 - x1| (x2 - x1)2

Substituting these into the Pythagorean theorem:

AB2 (x2 - x1)2 (y2 - y1)2

Taking the square root of both sides:

AB sqrt[(x2 - x1)2 (y2 - y1)2]

This is the distance formula for two-dimensional space.

Conclusion

The distance formula is a crucial tool in geometry and various mathematical applications. Its derivation from the Pythagorean theorem ensures its accuracy and reliability. Understanding this formula can help in solving a wide range of problems in mathematics and its applications in science and engineering.