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The Distance Formula

Consider the points (x₁, y₁) and (x₂, y₂) as shown in the figure below.

The distance between these points is the length of the hypotenuse of a right triangle as shown in the figure. The length of side a is y₂ - y₁ and the length of side b is x₂ - x₁. Using the Pythagorean theorem, we can write

d² = (x₂ - x₁)² + (y₂ - y₁)².

If we apply the even-root property and omit the negative square root (because the distance is positive), we can express this formula as follows.

Distance Formula

The distance d between (x₁, y₁) and (x₂, y₂) is given by the formula

Example

Using the distance formula

Find the length of the line segment with endpoints (-8, -10) and (6, -4).

Solution

Let (x₁, y₁) = (-8, -10) and (x₂, y₂) = (6, -4). Now substitute the appropriate values into the distance formula:

The exact length of the segment is .