Notes on Midpoint Formula - Page One
Name_________________________
The midpoint of a segment is the one point on a segment that is the same distance from both of
the endpoints of the segment.
M
A
B
For example, consider segment AB to the left.
It has endpoints named A and B.
The midpoint of the segment is labeled M. It is the
same distance from each of the endpoints.
same
distance
--------------------------------------------------------------------------------------------------------------------6
6
CD has been graphed on the coordinate
plane to the left.
C
4
D
C
N
4
D
2
2
-5
To the right, you can see that its midpoint
named point N has been plotted on the
segment.
5
-2
-4
-5
5
-2
-4
-6
-6
6
You can tell that point N is indeed the midpoint of the
segment by viewing the diagram to the left.
C
4
1
N
2
1
D
2
2
Note how two identical right triangles have been formed.
-5
5
-2
-4
-6
This means that the distance from C to N will be the same as
the distance from N to D. Therefore, point N is the midpoint
of CD .
--------------------------------------------------------------------------------------------------------------------So, how does one find the midpoint of a segment graphed on a
coordinate plane?
6
2
The segment from above has been recreated to the right, but now the
coordinates of the endpoints and midpoint are given.
-5
The y-coordinate of the midpoint is 4. This is the mean of the
y-coordinates of the endpoints (the mean of 3 and 5 is 4).
(5,3)
5
-2
The x-coordinate of the midpoint is 3. This is the mean (average) of
the x-coordinates of the endpoints (the mean of 1 and 5 is 3).
(1,5)
4
-4
-6
midpoint: (3,4)
Suppose one is given the coordinates of the endpoints of a line segment. Suppose they are
( x1 , y1 ) and ( x2 , y2 ) . In order to find the coordinates of the midpoint, the midpoint formula can
be used:
Coordinates of midpoint = (
x1  x2 y1  y2
,
)
2
2
--------------------------------------------------------------------------------------------------------------------Consider the segment graphed on the coordinate plane to the left.
What are the coordinates of the two endpoints?
6
4
Endpoint 1: (5, 2) Endpoint 2: (2, 4)
2
-5
5
x1 y1
-2
x2 y2
-4
-6
What are the coordinates of the midpoint of the segment?
Midpoint = (
x1  x2 y1  y2
5  2 2  4
,
) =(
,
) = ( 1.5,3)
2
2
2
2
When possible, it is always beneficial to check your answer by plotting the midpoint on the
coordinate plane to see if it is sensible. When (-1.5, 3) is plotted on the coordinate plane above,
you can see that it appears to be the midpoint of the segment.
--------------------------------------------------------------------------------------------------------------------Point A is located at (-2, 4), and point B is located at (1, 18). What are the coordinates of the
midpoint of AB ?
Midpoint = (
x1  x2 y1  y2
,
) =
2
2
--------------------------------------------------------------------------------------------------------------------Suppose a segment has one endpoint with coordinates of (3, 9). Also, suppose the segment has a
midpoint with coordinates of (10, 5). What are the coordinates of the other endpoint?
Note that the x-coordinate of one endpoint is given: x1  3 .
The x-coordinate of the other endpoint is unknown, so it will be known as x2 .
The x-coordinate of the midpoint is given - it is 10.
According to the midpoint formula:
x1  x2
 x-coordinate of the midpoint.
2
Notes on Midpoint Formula - Page Two
Name_________________________
However, since x1 and the x-coordinate of the midpoint are known, the expression can be
rewritten as:
3  x2
 10 .
2
Multiply both sides by 2 to eliminate the 2 in the denominator of the fraction on the left:
3  x2  20 .
Now, subtract 3 from both sides to find the x-coordinate of the other endpoint:
x2  17 .
Similarly, to find the y-coordinate of the other endpoint, use the formula:
y1  y2
 y-coordinate of the midpoint
2
Since y1  9 , and the y-coordinate of the midpoint is 5, the expression is rewritten as:
9  y2
5.
2
After multiplying both sides by 2 and subtracting 9 from both sides, y2  1 .
Thus, the coordinates of the other endpoint are (17,1) .
--------------------------------------------------------------------------------------------------------------------A segment has a midpoint located at (-2, -3) and an endpoint located at (-7, 2). What are the
coordinates of the other endpoint?
x1  x2
 x-coordinate of the midpoint
2
y1  y2
 y-coordinate of the midpoint
2
--------------------------------------------------------------------------------------------------------------------Suppose a segment has a midpoint at (10.5, -0.5) and an endpoint at (6, 11). What are the
coordinates of the other endpoint?
Extra Practice on Midpoint Formula
Name_________________________
For Questions 1-2, find the coordinates of the midpoint of the segment with the given endpoints.
1.
Endpoint 1: (13, 12)
Endpoint 2: (17, 5)
2.
Endpoint 1: (-10, -2)
Endpoint 2: (-1, -3)
3.
Point P is located at (-9, -5), and point Q is located at (-9, -8). What are the coordinates
of the midpoint of PQ ?
--------------------------------------------------------------------------------------------------------------------For Questions 4-5, consider the coordinate plane to the left.
6
4
A
2
-5
4.
What are the coordinates of the midpoint of AB ?
5.
What is the distance from point A to point B?
5
-2
B
-4
-6
--------------------------------------------------------------------------------------------------------------------6.
Suppose a segment has one endpoint with coordinates of (0, 11). Also, suppose the
segment has a midpoint with coordinates of (9, 10). What are the coordinates of the other
endpoint?
--------------------------------------------------------------------------------------------------------------------For Questions 7-8, find the coordinates of the other endpoint of the segment with the given
endpoint and midpoint.
7.
Endpoint 1: (13, 12)
Midpoint: (7.5, 5)
8.
Endpoint 1: (-10, -2)
Midpoint: (-2.5, -9.5)
--------------------------------------------------------------------------------------------------------------------9.
Point K is located at (4.2, 0.3) and point L is located at (5.8, -0.6). What is the location
of the midpoint of KL ?
--------------------------------------------------------------------------------------------------------------------10.
A segment on the coordinate plane has one endpoint located at (7, -7), and its midpoint
is located at (17.5, -0.5). What are the coordinates of the other endpoint of the segment?
--------------------------------------------------------------------------------------------------------------------11.
The midpoint of a segment is 11 units from one endpoint. How long is the segment?
--------------------------------------------------------------------------------------------------------------------1. (15, 8.5) 2. (-5.5,-2.5) 3. (-9, -6.5) 4. (-0.5, 1)
5. 5 units
6. (9,9)
7. (2, -2)
8. (5, -17)
9. (5, -0.15) 10. (28, 6)
11. 22 units