Distance Between Two Points

This topic is about the distance between two points in the two dimensional system.

Suppose we are given a point A with the co-ordinate (x_1, y_1) and a point B with the co-ordinate (x_2, y_2). We want to find the distance between point A and point B.

The distance from point A to point B is:

    \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

This distance formula is derived from the Pythagorean Theorem with the c^2 = a^2 + b^2 formula. (c is the longest side opposite to the right angle in the right angled triangle.)

Here is a diagram illustrating the geometric intuition behind the formula.

Source: http://www.helpingwithmath.com/images/geometry/distance-2points02.gif


Examples

Example One

Given the point A as (-1, 2) and B as (3, 0), the distance between point A and point B is:

    \[d = \sqrt{(3 - (-1))^2 + (0 - 2)^2} = \sqrt{(4)^2 + (- 2)^2} = \sqrt{16 + 4} = \sqrt{20} = \sqrt{4 \cdot 5} = 2 \sqrt{5}\]

Example Two

Suppose that the point A is (-2, 10) and the point B is (0, 4), the distance between point A and point B is:

    \[d = \sqrt{(0 - (-2))^2 + (4 - 10)^2} = \sqrt{(2)^2 + (- 6)^2} = \sqrt{4 + 36} = \sqrt{40} = \sqrt{4 \cdot 10} = \sqrt{4} \sqrt{10} = 2 \sqrt{10}\]

 

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