Transcript Distance Between Two Points

Let's find the distance between two points.
8
7 units apart
7
(-6,4)
6
5
4
3
2
1
(1,4)
So the distance from
(-6,4) to (1,4) is 7.
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
-2
-3
-4
-5
-6
-7
If the points are located horizontally from each other,
the y coordinates will be the same. You can look to
see how far apart the x coordinates are.
What coordinate will be the same if the points
are located vertically from each other?
(-6,4)
7 units apart
8
7
6
5
4
3
2
1
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
-2
-3
-4
-5
-6
-7
(-6,-3)
So the distance from
(-6,4) to (-6,-3) is 7.
If the points are located vertically from each other,
the x coordinates will be the same. You can look
to see how far apart the y coordinates are.
But what are we going to do if the points are not located either
horizontally or vertically to find the distance between them?
Let's start by finding the
distance from (0,0) to (4,3)
8
7
6
5
4
3
2
1
a b  c
2
3
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
-2
-3
-4
-5
-6
-7
The Pythagorean
Theorem will help us
find the hypotenuse
2
4 3  c
2
5?
4
2
2
2
16  9  c
2
c5
So the distance between (0,0)
and (4,3) is 5 units.
This triangle measures 4 units by 3
Let's add some lines and units on the sides. If we find the
hypotenuse, we'll have the distance
make a right triangle.
from (0,0) to (4,3)
Now let's generalize this method to come up with a formula so
we don't have to make a graph and triangle every time.
8
7
6
2 5
4
3
2
11
Let's start by finding the
distance from (x1,y1) to (x2,y )
a b  c
2
(x2,y2)
?
2
y2 – y1
(x ,y1)
x -x
-7 -6 -5 -4 -3 -2 -1 0 1 2 32 4 5 16 7 8
-2
-3
2
-4
2
1
-5
-6
-7
x
Again the Pythagorean
Theorem will help us
find the hypotenuse
Let's add some lines and
make a right triangle.


 x   y2  y1  c 2
2
Solving for c gives us:
c
x2  x1    y2  y1 
2
2
This is called the distance formula
2
Let's use it to find the distance between (3, -5) and (-1,4)
Plug these values in
the distance formula
2
c   4   9 
2
CAUTION!
(x2,y2)
x-12  x31    y42  -5y1 
c
2
(x1,y1)
2
means
approximately
equal to
 16  81  97  9.8
found with a
calculator
Don't forget the order of operations!
You must do the brackets first then
powers (square the numbers) and then add
together BEFORE you can square root
Acknowledgement
I wish to thank Shawna Haider from Salt Lake Community College, Utah
USA for her hard work in creating this PowerPoint.
www.slcc.edu
Shawna has kindly given permission for this resource to be downloaded
from www.mathxtc.com and for it to be modified to suit the Western
Australian Mathematics Curriculum.
Stephen Corcoran
Head of Mathematics
St Stephen’s School – Carramar
www.ststephens.wa.edu.au