If you have two points on a graph, such as (2, 3) and (2, 6), you
can find the distance between them simply by counting units on the graph,
as they lie in a vertical line on the graph.
The distance between these two points is three. You could find the same
thing with a horizontal line, simply by counting the number of units between
the two points.
However, most sets of points do not lie on a vertical or horizontal line.
For example:
Counting the units between these two lines is impossible. So mathematicians
have developed a formula using the Pythagorean theorem to find the distance
between two points. That formula is
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Here is how the distance formula is derived.
If you are trying to find the distance between these two points,
Point A (red), (2,6) and Point B (blue), (4,3), notice that they
form a right triangle with Point C (purple), (2,3).
In order to understand the derivation, you need to change the points
into symbols. Point A will become (x1, y1) and point B will become
(x2, y2). Notice that the coordinates of Point C correspond as follows
(x1, y2).
Now, label the distance between Point A and Point B d. This is
what we are trying to find.
You can find the distance between A and C by finding the difference
between their y coordinates, since the x coordinate is two for both
points. You would be finding the absolute value, as you cannot have
a negative distance. So the distance between A and C would be shown
as 
Conversely, finding the distance between Point B and Point C would
involve finding the difference of the x coordinates, as follows:
Now, we are still looking for d, which you will notice is the hypotenuses
of the right triangle ABC. The Pythagorean theorem says that a2
+ b2 = c2
In our example, the hypotenuse is d, so we substitute
a2 + b2 = d2
And since a and b stand for the distance of the two legs, we can
substitute:
(
)2 +(
)2 = d2
Since squaring an absolute value makes the absolute value no longer
necessary, we can simplify this to (x1 – x2)2 +
(y1 – y2)2 = d2
To solve for d we take the square root of both sides, giving us
d =
-- the Distance Formula!
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Once you have found the set of points you want to find the distance of,
you need to label them as point 1 and point 2. This is very important,
as it helps you keep your points straight in the equation.
In this graph, the two points are (2,6) and (4,3). Label them as follows:
Point 1--(2,6) Point 2--(4,3).
Now, put the points into the equation. The first point will be x1
and y1. It is very important to keep them in the right order!
= 
= 
=  --this
does not simplify, so it is our answer.
Example 2
Find the distance between (-3,4) and (1,7)
Start by labeling: Point 1 - (-3,4) and Point 2 - (1,7)
Fill in the equation:
= 
= 
= 
= 5
Notice we only use the positive root. This is because we are measuring
a distance, which cannot be negative.
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Example 3
Point 1 - (-5,-2)
Point 2 - (-4,-7)
Notice that all of the points are negative. This will be important
in figuring the problem.
Now, two negatives in a row, according to the rules of multiplication,
make a positive:
= 
= 
= 
= 
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