Distance Formula.notebook
November 03, 2013
Distance Formula
y
6
What is the distance from A
5
4
to B?
A
B
3
2
1
­6
units
­5
­4
­3
­2
­1
0
­1
x
1
2
3
4
5
6
­2
­3
­4
­5
­6
1
Distance Formula.notebook
November 03, 2013
y
6
Example:
5
4
3
2
1
­6
­5
­4
­3
­2
­1
0
­1
x
1
2
3
4
5
6
­2
­3
C
­4
D
­5
­6
To calculate the horizonal distance between two
points, subtract the x co-ordinates:
2
Distance Formula.notebook
November 03, 2013
What is the distance between points
and
?
Points that are in line horizontally have the same y
coordinates.
3
Distance Formula.notebook
November 03, 2013
What is the distance from M to N?
y
units
8
7
6
5
4
3
2
1
M
N
x
­8 ­7 ­6 ­5 ­4 ­3 ­2 ­1 ­10 1 2 3 4 5 6 7 8
­2
­3
­4
­5
­6
­7
­8
4
Distance Formula.notebook
November 03, 2013
y
10
9
8
7
6
5
4
3
2
1
units
­10
­8
­6
­4
­2
0
­2
­3
­4
­5
­6
­7
­8
­9
­10
M
x
2
4
6
8
10
N
To find the vertical distance between two points,
subtract the y co-ordinates:
5
Distance Formula.notebook
What is the distance between points
November 03, 2013
and
?
Points that are in line vertically have the same x coordinates.
6
Distance Formula.notebook
November 03, 2013
For oblique (slanted) lines:
y
10
9
8
7
6
5
4
3
2
1
Find the distance
from J to K.
­10
­8
­6
­4
­2
0
J
x
2
4
6
8
10
­2
­3
­4 K
­5
­6
­7
­8
­9
­10
Hint: Think of
Pythagoras' Theorem
7
Distance Formula.notebook
November 03, 2013
On the Cartesian plane, we
y
10
9
8
7
6
5
4
3
2
1
can make a right triangle
whose hypotenuse is the
segment JK.
By calculating the vertical
­10
­8
­6
­4
­2
0
2
­2
(1 ,
­3
­3)
­4 K
­5
­6
­7
­8
­9
­10
(8 ,
6)
J
x
4
6
8
10
and horizontal distances
(the lengths of the legs of the triangle) we can use
Pythagoras' theorem to find d(J, K ).
8
Distance Formula.notebook
November 03, 2013
Pythagoras' Theorem
y
10
9
8
7
6
5
4
3
2
1
­10
­8
­6
­4
­2
0
2
­2
(1 ,
­3
­3)
­4 K
­5
­6
­7
­8
­9
­10
(8 ,
6)
J
c
4
b
6
8
x
10
a
d(J, K) = 11.4 units
9
Distance Formula.notebook
November 03, 2013
y
10
9
8
7
6
5
4
3
2
1
Calculate d(P, Q)
P
­10
­8
­6
­4
­2
­10
­2
­3
­4
­5
­6
­7
­8
­9
­10
Q
x
2
4
6
8
10
10
Distance Formula.notebook
November 03, 2013
Calculate d(G, H)
y
10
9
8
7
6
5
4
3
2
1
1. Find the horizontal
distance
.
­10
2. Find the vertical
distance
.
­8
­6
­4
­2
0
G
x
2
4
6
­2
­3
­4
­5
­6
­7
­8
­9
­10
8
10
H
3. Calculate the distance
between the points using Pythagoras' theorem.
11
Distance Formula.notebook
November 03, 2013
1.
y
10
9
8
7
6
5
4
3
2
1
2.
3.
­10
­8
­6
­4
­2
0
­2
­3
­4
­5
­6
­7
­8
­9
­10
G(1 ,
x
1)
2
4
6
8
10
(9 ,
­5)
H
d(G,H) = 10 units
12
Distance Formula.notebook
November 03, 2013
These steps can be condensed into a single formula
called the ...
Distance Formula.
13
Distance Formula.notebook
November 03, 2013
Example: Calculate .
y
Note:
A
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
­14 ­12 ­10 ­8 ­6 ­4 ­2 ­10
­2
­3
­4
­5
B
x
2
4
6
8 10 12 14
14
Distance Formula.notebook
November 03, 2013
Example: Calculate
.
y
10
9
8
7
6
5
4
3
2
1
­10
­8
­6
­4
­2
­10
­2
­3
­4
­5
­6
­7
­8
­9
­10
F
x
2
4
6
8
10
L
15