The Distance Formula
Consider the points (x_{1}, y_{1}) and (x_{2}, y_{2}) 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 y_{2}  y_{1} and the length of side b is
x_{2}  x_{1}. Using the
Pythagorean theorem, we can write
d^{2} = (x_{2}  x_{1})^{2} + (y_{2}
 y_{1})^{2}.
If we apply the evenroot 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 (x_{1}, y_{1}) and (x_{2}, y_{2}) 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 (x_{1}, y_{1}) = (8, 10) and (x_{2}, y_{2})
= (6, 4). Now substitute the appropriate
values into the distance formula:
The exact length of the segment is
.
