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1. Distance, Midpoint,
Pythagorean Theorem

2. Distance Formula
• Distance formula—used to measure the
distance between between two endpoints of
a line segment (on a graph).
d x x y y
   
( ) ( )
1 2
2
1 2
2
• x1
and y1
are the coordinates of the first point
• x2
and y2
are the coordinates of the second point

3. Distance Formula
• The distance between two points A(x1 , y1) and
B(x1 , y1) is
2 2
2 1 2 1
( ) ( ) .
d x x y y
   

4. Finding Distance
• What is the distance between U(-7 , 5) and
V(4 , -3)? Round to the nearest tenth?
2 2
2 1 2 1
( ) ( )
d x x y y
   
2 2
(4 ( 7)) ( 3 5)
d      
2 2
(11) ( 8)
d   
121 64
 
185
 13.6


5. Distance Formula examples
• Find the distance between the points (1, 2)
and (–2, –2).
• Your video game uses a coordinate grid
system for location. There is an enemy ship
at (7, –3). You are at (–8, –3). If one grid
unit equals 10 miles, how far away is the
enemy ship?

6. Midpoint Formula
• Midpoint formula—used to find the
midpoint of a line segment. (It will always
be in the form of a point (x, y).)





 


2
,
2
2
1
2
1 y
y
x
x
M
• x1
and x2
are the x-coordinates of the points
• y1
and y2
are the y-coordinates of the points

7. Finding the Midpoint
• Segment AB has endpoints at -4 and 9. What is the
coordinate of its midpoint?
2
a b
M


4 9
2
M
 

5
2
M  2.5


8. Finding the Midpoint
• Segment EF has endpoints E (7 , 5) and
F (2 , -4). What are the coordinates of its midpoint
M?
1 2 1 2
,
2 2
x x y y
M
 
 
 
 
7 2 5 ( 4)
,
2 2
M
  
 
 
 
9 1
,
2 2
M
 
 
 
(4.5,0.5)
M

9. Finding an Endpoint
• The midpoint of segment CD is M(-2 , 1). One
endpoint is C (-5 , 7). What are the coordinates if
the other endpoint D?
2 2
5 7
( 2,1) ,
2 2
x y
  
 
  
 
2
5
2
2
x
 
 
2
4 5 x
  
2
1 x

2
7
1
2
y


2
2 7 y
 
2
5 y
  (1, 5)
D 

10. 1.8 Midpoint and Distance in the Coordinate
Plane
• You can use formulas to find the midpoint and the length of
any segment in the coordinate plane.
Number Line Coordinate Plane
2
a b
M


1 2 1 2
,
2 2
x x y y
M
 
 
 
 

11. Midpoint Formula
• Find the midpoint of the segment given the
endpoints (5, 7) and (13, 1).
• What is the midpoint of the line segment
with endpoints (–3, –3) and (7, 3)?
• Line segment CD has a midpoint at (1, 2). If
endpoint C is located at (–5, 3), find the
ordered pair represented the other endpoint D.

12. Pythagorean Theorem
• Pythagorean Theorem—In a right
triangle, the sum of the squares of the two
legs equals the hypotenuse squared.
• a2
+ b2
= c2
a and b are legs
c is the hypotenuse

13. Pythagorean Theorem
leg
leg
hypotenuse
The hypotenuse is always the longest side of a right
triangle and is always opposite the right angle.

14. Pythagorean Theorem
• What is the value of the missing side?
5
12

15. Pythagorean Theorem
• What is the value of the missing side?
9 15

16. Pythagorean Theorem
• The perimeter of a square is 36 inches.
What is the length of its diagonal?

17. Homework
• Pg. 552 (#10, 11, 14, 15, 22-27)
• Pg. 557 (#10-18 find distance AND midpoint)