The Distance between Two Points

Understanding The Concept

In Rectangular Cartesian Coordinates, the distance between two points can be calculated using a special formula derived from the Pythagorean theorem: the Distance Formula.
The two points to consider generally have coordinates: point P(x1, y1) and point Q(x2, y2).
The distance, denoted as d(P, Q), between these two points P and Q is given by the formula:

d(P, Q) = sqrt[(x2-x1)^2 + (y2-y1)^2]

where sqrt is the square root function.

The inside of the square root calculates the sum of the squares of the differences in x-coordinates and y-coordinates of points P and Q.
The expression (x2-x1) gives the horizontal distance (along the x-axis) between P and Q; (y2-y1) gives the vertical distance (along the y-axis).
Squaring these distances and adding them gives us the square of the direct distance between P and Q (d(P, Q)^2).
The square root of this results in the direct distance, d(P, Q).

For instance, calculate the distance between points A(2,3) and B(5,7): Substitute values in the formula: d(A, B) = sqrt[(5-2)^2 + (7-3)^2] = sqrt[9 + 16] = sqrt[25] = 5
When you have a point and want to find another point a given distance away, rewrite the formula as an equation for the coordinates of the second point and solve for those values.

The Distance Formula is a vital tool for these coordinates in pure mathematics.
By understanding the logic behind the formula, not just memorising it, you give yourself the best chance of being able to apply it successfully in different situations.