



What
else could we do with two points? 





We
could find the distance between them! 





Let's
find the distance between 


our
old friends from the last page ... 











So
how do we do it? 





We
use something called the Pythagorean Theorem 


(What
a name, eh?) 





Anyway,
it works when you have a triangle 


with
a 90 degree angle (called a right triangle). 





The
deal is, in a right triangle, 


if
you take the lengths of the two shorter sides 


multiply
them times themselves and add that together 


you
get the same answer as you do 


when
you take the longest side of the triangle 


and
multiply it times itself. 





In
math talk ... 











Oh
yeah?, 


Well
I didn't see any triangles on the graph. 





Well
then you're not looking hard enough.



It's
there ... really. 


One
of the lines of the triangle, 


is
the line between the two points. 











and
a line that runs straight down 


from
the point on the right ... 











and
a line that runs straight across 


from
the point on the left. 











If
you put them together, you get ... 











Since
one of these last two lines runs exactly up and down 


and
the other one runs exactly left and right 


where
they meet, they make a 90 degree angle. 











We
also know that the point where the two lines meet 


has
the X value of the point on the right 


and
the Y value of the point on the left. 


Study
the picture until you understand why. 





Now
we can find the lengths of the two shorter sides. 


The
one that runs from left to right 


goes
from the point (1,2) to the point (5,2) 


For
both of these points, Y = 2. 


That
means the distance is the difference 


between
the X coordinates ... 











We
can use this trick to find the distance 


between
the two points on the right side. 





The
X values of these two points are both 5, 


only
the Y values are different. 


The
distance between these points is that difference. 











Now
we know the lengths we need 


to
use that Pythagorean Theorem thing. 





It
will let us figure out the length 


of
the long side of the triangle. 


Which
just happens to be the distance 


between
the two points! 

















This
time, we don't have to worry about that +/ thing 


that
you usually get with a root. 


That's
because we're talking about a distance here. 


And
this distance is not going to be negative number. 





So
the distance between the points (1,2) and (5,3) 


is
about 4.124. 





copyright 2005 Bruce Kirkpatrick 
