1. The document discusses distance and midpoints between two points. It provides the distance formula and explains how to find the distance and midpoint between two points given their x and y coordinates. 2. It gives examples of using the distance formula to calculate the distance between points like (-3,2) and (1,-1), which is √25 or 5 units. 3. The midpoint between two points is calculated by taking the average of the x-coordinates and the average of the y-coordinates. For points (7,2) and (-3,6), the midpoint is (2,4).

1. 1.3 Distance and
Midpoint

2. A B
The distance from point A
to point B
is 4 units.
Count spaces
not lines.

3. E
D
The distance from point D
to point E
can be found
with
a2 + b2 = c2 3 c
4
32 + 42 = c2
5 = c
5

4. You can also use the
distance
formula.
(-3,2)
(1,-1)
Use ordered
pairs.
E
D

5. The distance between
two points (x1,y1) and
(x2,y2) is
d = x2 - x1
( )
2
+ y2 - y1
( )
2

6. y1
y2
x1 x2
(x1,y1)
(x2,y2)
lx2-x1l
ly2-y1l
a2
b2
d
c2

7. d = x2 - x1
( )
2
+ y2 - y1
( )
2
a 2 + b2
c = √
Distance formula is based
on Pythagorean Theorem.

8. Find distance between
(-3,2) and ( 1, -1 )
(x1, y1) (x2, y2)
_______________
d = √( 1++3)2 + ( - 1- 2 )2
d = x2 - x1
( )
2
+ y2 - y1
( )
2
1 -3 -1 2

9. Find distance between
(-3,2) and ( 1, -1 )
d = √( 1++ 3)2 + ( - 1-2 )2
d = √( 4 )2 + ( - 3 )2
d = √ 16 + 9
d = √25

10. Find distance between
(5, 1) and ( 2, -6 )
(x1, y1) (x2, y2)
d = x2 - x1
( )
2
+ y2 - y1
( )
2
( 2 - 5)2 + ( -6 - 1 )2
d = x2 - x1
( )
2
+ y2 - y1
( )
2

11. d = x2 - x1
( )
2
+ y2 - y1
( )
2
d = x2 - x1
( )
2
+ y2 - y1
( )
2
( 2 - 5)2 + ( -6 - 1 )2
d =√
d =√
d =√
(- 3) 2 + ( - 7 )2
9 + 49
58 = 7.6
careful !
this is not
- 32

12. (-3)2 = 9
-32 is NOT the same
Don’t forget
the ( ) !!!

13. A B
The distance from point A
to point B
is 4 units.
The
MIDPOINT
is 1/2 way.

14. For any 2 points (x1, y1)
and (x2, y2) the midpoint is
x1 + x2
2
,
y1 + y2
2
æ
è
ç
ö
ø
÷
(x1,y1)
(x2,y2)

15. Find midpoint
(7, 2) and ( -3, 6 )
x1 + x2
2
,
y1 + y2
2
æ
è
ç
ö
ø
÷
(x1, y1) (x2, y2)

16. x1 + x2
2
,
y1 + y2
2
æ
è
ç
ö
ø
÷
x1 + x2
2
,
y1 + y2
2
æ
è
ç
ö
ø
÷
-3 + 7 , 6 + 2
2 2
x1 + x2
2
,
y1 + y2
2
æ
è
ç
ö
ø
÷
4 , 8
2 2 ( 2, 4)
(7, 2) and ( -3, 6 )

17. Segment AB, has one
endpoint A (-1,5). The
midpoint is C( 7,3). Find the
other endpoint B.
A
C B

18. Endpoint A (-1,5). (x1,y1)
Midpoint C( 7,3). Find endpoint B.
x1 + x2
2
,
y1 + y2
2
æ
è
ç
ö
ø
÷ = (7,3)
x1 + x2
2
= 7
y1 + y2
2
= 3
-1 5
x2= 15 y2= 1
(15,1)

19. What kind of triangle is this?
Find the lengths of the sides.
d
d d
3 = sides is
equilateral.
2 = sides is
isosceles.
no = sides
scalene. d = x2 - x1
( )
2
+ y2 - y1
( )
2

20. The pessimist sees difficulty
in every opportunity.
The optimist sees
the opportunity in
every difficulty.