Distance Formula

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Theorem

The length of the line segment AB, which joins A (x₁, y₁) and B (x₂, y₂) is given by

Proof

Let A (x₁, y₁) and B (x₂, y₂) be two points in the plane.

Let d = distance between the points A and B.

Draw AL and BM perpendicular to x-axis (parallel to y-axis).

Draw AC perpendicular to BM to cut BM at C.

In the figure,

OL = x₁, OM = x₂ [AC = LM = OM - OL = x₂ - x₁]

MB = y₂, MC = LA = y₁ [CB = MB - MC = y₂ - y₁]

From the right-angled DACB,

Note:

i) If the points A and B lie on the x-axis, then the ordinates of A and B are zeros.

i.e., A (x₁, 0), B (x₂,0)

ii) If the points A and B lie on the y-axis, then the abscissae of A and B are zeros.

i.e., A (0,y₁) and B (0,y₂)

iii) Distance of any point A (x, y) from the origin

Example:

Find the distance between the following pair of points:

A (1,2) and B (4,5).

Suggested answer:

Using the distance formula, we have