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.


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