Transcript Distance and Midpoints

Distance and Midpoints
Objective:
(1)To find the distance between two points
(2) To find the midpoint of a segment
Definitions
• Midpoint: The points halfway between the
endpoints of a segment.
• Distance Formula: A formula used to find the
distance between two points on a coordinate
plane.
• Segment Bisector: A segment, line, or plane
that intersects a segment at its midpoint.
Midpoint
• To find the midpoint
along the number line,
add both numbers and
divide by 2.
ab
2
A B C D E F G H I J
-6 -4 -2 0 2 4 6 8 10 12
Find the midpoint of BH
48
4
 2
2
2
The coordinate of the
midpoint is 2.
E is the midpoint.
More Midpoint
• For the midpoint on a
coordinate plane, the
formula is:
 x1  x2 y1  y2 
M
,

2 
 2
B(-1,7)
A(-8,1)
  8  (1) 1  7 
,


2
2 

 9 8
, 

 2 2
This is the midpoint.
1 

  4 ,4 
2 

Finding the endpoint of a segment
• We’re still going to use
the Midpoint Formula:
 x1  x2 y1  y2 
M
,

2 
 2
• But the there will be a
few unknowns:
Find the coordinates for X
if M(5,-1) is the
midpoint and the other
endpoint has
coordinates Y(8,-3)
x1  x2
•
helps us find
2
the x-coordinate of the
endpoint.
Finding the endpoint of a segment
8  x2
5
2
Multiply both sides by 2 to eliminate the denominator
 8  x2 
2
  25
 2 
y1  y2
2
 3  y2
 1
2

8  x2  10
-8
x2 = 2
-8
Subtract 8 from both sides
This is the x-coordinate of
the other endpoint
This helps us find
the y-coordinate of
the midpoint
  3  y2 
2 2   21
Finding the endpoint of a segment
 3  y2  2
+3
y2 = 1
+3
This is the y-coordinate of the
endpoint
The coordinate of the other
endpoint is X(2,1).
Finding the value of a variable
M is the midpoint of AB.
Find the value of x:
Since M is a midpoint, that
means that AM=MB which
means
3x – 5 = x + 9
-x
-x
2x – 5 = 9
+5 +5
2x = 14
 A 3x - 5

M
x+9
B
2x = 14
2
x=7
2
Distance
• Remember:
AB means the length of
AB
To find the distance on the
number line, take the
absolute value of the
difference of the
coordinates.
a – b
A B C D E F G H I J
-6 -4 -2 0 2 4 6 8 10 12
Find CJ
-2 -12=-14= 14
CJ = 14
Find EA
2 – (-6)
=2+6
=8
=8
EA = 8
More Distance
The distance between two
points in the coordinate
plane is found by using
the following formula:
A(-3,1)
B(4,-2)
d  [4  (3)] 2  (2  1) 2
d  ( x2  x1 ) 2  ( y2  y1 ) 2
d  (7) 2  ( 3) 2
d  49 9
d  58
d  7.6