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

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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