Distance Formula





Basic Distance Formula:

The basic distance formula states that:

The distance “d” between two points A(x1,y1) and B(x2,y2) can be calculated as:

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

 

Using this Distance Formula of coordinate geometry we can establish fundamental trigonometric formulae for general angles in a very elegant way. So we shall now prove or derive this formula:

 

Derivation or Proof of Distance Formula:

In the adjoining figure , “d” is the distance between two points P(x1 , y1) and Q(x2 , y2).

basic distance formula

basic distance formula

Now let us draw “PL” perpendicular to “OX” and “PR” perpendicular to “QM”.

Now , PR = OM - OL

= x_2 - x_1

 

And Similarly:

QR = QM - RM

= y_2 - y_1

 

And now in the right angle triangle PQR applying Pythagorean  Theorem:

PQ^2 = PR^2 + RQ^2

So , d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2

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

This is the basic distance formula and can be used to calculate to find the distance between two points P , Q if their co-ordinates ( x1 , y1 , x2 , y2) are known.

 



Related posts:

  1. Average velocity and Average speed. Average velocity: Average velocity is the measure of the average...
  2. Unit Vector Unit Vector. What is unit vector? What are standard unit...
  3. Trigonometric functions of negative angles Trigonometric functions of negative angles. How to find trigonometric functions...
  4. Constant acceleration & Constant acceleration equations Constant acceleration. And Constant Constant acceleration equation. Derivation of constant...
  5. Pythagorian Identities Fundamental Pythagorian identity of trigonometry and other basic trigonometric formulas...