Distance Between Two Points

The distance between any two points P(x₁, y₁) and Q(x₂, y₂) in the plane is the length of the line segment PQ.

From P, Q draw PL and QM perpendicular on the x-axis and PR perpendicular on QM. Then,

OL = x₁, OM = x₂, PL = y₁ and QM = y₂

PR = LM = OM - OL = x₂ - x₁

QR = QM - RM = QM - PL = y₂ - y₁

Since PQR is a right angled triangle, by Pythagoras Theorem

$$ PQ^2 = PR^2 + QR^2 $$

$$ PQ^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2 $$

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

$$ \text{ Distance between two points} = \sqrt{\text{(difference of abscissae)}^2 + \text{(difference of ordinates)}^2} $$

Distance of Point from Origin

The distance of the point (x₁, y₁) from the origin (0, 0) is

$$ = \sqrt{x_1^2 + x_2^2} $$