Today we will explore the Essential Question, “What is the method for
finding the midpoint between two given points on a coordinate plane?”
The point halfway between the endpoints of a given segment is
called the midpoint.
A midpoint divides a line segment into two equal parts.
To find the midpoint of a line segment on a coordinate plane, find the
average of the x-coordinates and the average of the y-coordinates.
These averages will be the x and y coordinates of the midpoint of the
given segment.
The concept of finding these averages
can be written as a formula and it is
written on the FCAT Mathematics
Reference Sheet as shown in the box.
Midpoint between two points
P1( x1 , y 1) and P2 ( x 2 , y 2 ) :
 x2  x1 y2  y1 
,


2 
 2
Example 1: Find the midpoint of a segment with endpoints at
 5,1 and  7,9 
Name the coordinates of
 7, 9 as ( x 2 , y 2 )
.
 5,1 as ( x 1 , y 1 )
and the coordinates of
and substitute into the formula as follows:
 x2  x 1 y2  y1   7  ( 5) 9  1    2 , 10    1,5  .
,
,


  2 2 
2
2
2
2

 

Therefore, the midpoint of the segment is (1,5).
Example 2: Point Z is the midpoint of a segment with endpoints at
X and Y with integer coordinates as shown in the diagram. Find the
coordinates of point Z.
y
It is necessary to first find the
coordinates of points X and Y.
Point X is located at
10
 7, 6
 7, 6 
8
6
X
4
and point Y is located at  6,4  .
Name the coordinates of  7, 6  as
2
x
-10
( x 1 , y 1)
and the coordinates of  6. , 4  as ( x 2 , y 2 )
and substitute into the formula as follows:
-8 -6
-4
-2
0
-2
-4
-6
-8
-10
 x2  x 1 y2  y1   6  ( 7) 4  6   1 2   1 
,
,


   ,     ,1 
2
2  
2
2   2 2  2 

 1 
Therefore, point Z is located at   , 1  .
 2 
2
4
6
8
10
 6,4
Y
y
Guided Practice Problems:
10
1. Find the coordinates of the midpoint of
a segment with endpoints at W and Y.
(2, 9)
6
4
W is located at (2, 9). Y is located at (8, 3).
Name (2, 9) as  x1 , y1  and (8, 3) as  x2 , y2  .
 x2  x 1 y2  y1   8  2 3  9 
,
,



2
2   2
2 

W
8
Y
(3, 1) (8, 3)
2
Z
-10
-8
-6
-4
-2
0
2
x
4
6
8
10
-2
X
(-3,-3)
-4
-6
 10 12 
 ,    5,6  .
 2 2 
-8
-10
2. Find the coordinates of the midpoint of
a segment with endpoints at X and Y.
X is located at (-3, -3).
3. Find the coordinates of
the midpoint of a segment
Y is located at (8, 3). with endpoints at W and Z.
Name (-3, -3) as  x1 , y1  and (8, 3) as  x2 , y2 .
 x2  x 1 y2  y1   8  ( 3) 3  ( 3) 
,
,



2
2  
2
2


5 0
 ,    2.5,0  .
 2 2
W is (2, 9).
x , y 
1
1
Z is (3, 1).
x , y 
2
2
 x2  x 1 y2  y1   3  2 1  9 
,
,



2
2
2
2

 

 5 10 
 ,    2.5,5  .
2 2 
4. If the coordinates of the endpoints of a segment are located at
(1, 5) and  4,9  , find the coordinates of the midpoint.
Name the coordinates of
 4, 9 as ( x 2 , y 2 )
 1, 5  as ( x 1 , y 1 )
and the coordinates of
and substitute into the formula as follows:
 x2  x 1 y2  y1   4  1 9  ( 5)    5 , 4    5 ,2  .
,
,


  2 2   2 
2
2
2
2

 

5 
Therefore, the midpoint of the segment is  2 ,2  .


NOTE: The fraction
5
2
can be written as 2.5.
Independent Practice:
1. If the coordinates of the endpoints of a segment are located at
 0,8 and  6,1 , find the coordinates of the midpoint.
Name the coordinates of  0,8  as
( x 1 , y 1 ) and  6, 1 as ( x 2 , y 2 )
9
 x2  x 1 y2  y1   6  0 1  8   6 9  

,


3,
,

,
.

 
  2 2 
2
2
2   2
2  
 

y
2. A monorail at an amusement park has
stops at points W and Z as shown on the
coordinate plane. A new stop needs to be
inserted at the midpoint of the segment with
endpoints at W and Z. What are the
coordinates of the location of the new stop?
10
8
W
Name (-6, 6) as  x , y  and (10, -9) as  x , y  .
1
1
 x2  x 1 y2  y1   10  ( 6) 9  6 
,
,



2
2  
2
2 

3
 4 3  
,

2,


 
.
2
2
2

 

2
2
6
4
2
x
-10
W is located at (-6, 6). Z is located at (10, -9).
.
-8
-6
-4
-2
0
2
4
6
8
10
-2
-4
-6
-8
-10
Z