5.7 ­ The Distance Formula
What's the distance between the two points?
1{
{
a2 + b2 = c2
(9)2 + (1)2 = c2
81 + 1 = c2
82 = c2
c = 9.1
9
The Distance Formula:
to find the distance between (x1, y1) and (x2, y2)
5.7 ­ The Distance Formula
Pythagorean Theorem
Rewriting to solve for side c in one step
Pythagorean Theorem
a2 + b2 = c2
Rewritten
c2 = a2 + b2
Solving for side c
One step to solve for side c
c2 = c = How we get the distance formula:
(x2 ­ x1) same as finding length of side a
(y2 ­ y1) same as finding length of side b
5.7 ­ The Distance Formula
(x1, y1) (­5, 1)
4 ­ (­5)
9
(4, 0) (x , y )
2
2
0 ­ 1
­1
81 + 1
82
9.1
Using the distance formula:
Step 1) Find the two ordered pairs you are given
Step 2) Choose one ordered pair to be (x1, y1) Step 2)
Label the other ordered par to be (x2, y2)
Step 3)
Plug your numbers into the distance formula and begin simplifying
5.7 ­ The Distance Formula
Find the distance between the two points below:
Round your answer to the nearest tenth, if necessary.
(x1, y1)(­5, 5)
1 ­ (­5)
6
(1, ­2)(x , y )
2
2
­2 ­ 5
­7
36 + 49
85
9.21954.... d = 9.2
Find the distance between the two points below:
Round your answer to the nearest tenth, if necessary.
(­4, ­2)
(x1, y1)
(4, 3) (x , y )
2
4 ­ (­4)
3 ­ (­2)
8
5
2
64 + 25
89
9.43398.... d = 9.4
5.7 ­ The Distance Formula
Find the distance between the two points below:
Round your answer to the nearest tenth, if necessary.
(−6, −10) (2, 4)
(x1, y1)
(x2, y2)
2 ­ (­6)
4 ­ (­10)
8
14
64 + 196
260
16.12451..... = 16.1
Find the distance between the two points below:
Round your answer to the nearest tenth, if necessary.
(1, −9) (6, ­6)
(x1, y1)
(x2, y2)
6 ­ (1)
­6 ­ (­9)
5
3
25 + 9
34
5.830951.... d=5.8
5.7 ­ The Distance Formula
Chicago, Illinois, has a longitude of 88°W and a latitude of 42°N. Indianapolis, Indiana, has a longitude of 86°W and a latitude of 40°N. At this longitude/latitude, each degree is about 53 miles. Find the distance between Chicago and Indianapolis. (x1, y1) (86, 40)
88 ­ 86
2
4 + 4
8
(88, 42)(x2, y2)
42 ­ 40
2
2.828427...
x 53 (miles per unit)
149.9066... = about 150 miles