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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 .

 

 
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