When
working with Coordinate Geometry, there are many ways to find distances
(lengths) of line segments on graph paper.
Let's examine some of the possibilities:

Whenever
the segments are horizontal or vertical, the length can be
obtained by counting.
When
we need to find the length (distance) of a segment such as
we simply COUNT
the distance from point A to point B.
(AB = 7)
We can use this same
counting approach for
.
(CD = 3)

Unfortunately,
this counting approach does NOT work for
which is a diagonal segment.

When
working with diagonal segments, the Pythagorean theorem can be
used to determine the length.
Notice
how a right triangle was formed with
as the hypotenuse. The legs of the right triangle are
"on" the graph paper and their lengths can be
counted. Using the Pythagorean Theorem, we know that:



Use
this method of working with the Pythagorean Theorem whenever you
forget the Distance Formula! 

When working with diagonal segments,
use the
Distance Formula
to determine the length.
The
advantage of the Distance Formula is that you do not need to draw
a picture to find the answer. All you need to know are the
coordinates of the endpoints of the segment.
It
doesn't matter which point you start with. Just start with
the same point for reading both the x and y coordinates.

The
Distance Formula is really just a coordinate geometry way of
writing the Pythagorean Theorem. If you cannot remember the
Distance Formula, you can always draw a graph and use the
Pythagorean Theorem as was done in Method 2. 
The
Distance Formula can be used to find the lengths of all forms of
line segments: horizontal, vertical and
diagonal.
Deriving the Distance
Formula:

