The
point halfway between the endpoints of a line segment is called the midpoint.
A midpoint divides a line segment into two equal segments.
By definition,
a midpoint of a line segment
is the point on that line segment that divides the segment two congruent
segments.
In Coordinate
Geometry, there are several ways to determine the midpoint of a line
segment.
If the
line segments
are vertical or horizontal, you may find the midpoint by simply dividing the
length of the segment by 2 and counting that value from either of the
endpoints.

Find
the midpoints and
.
AB
is 8 (by counting). The midpoint is 4 units from either
endpoint. On the graph, this point is (1,4).
CD is 3 (by counting). The midpoint is 1.5 units
from either endpoint. On the graph, this point is (2,1.5)

If the
line segments
are diagonally positioned, more thought must be paid to the solution. When
you are finding the coordinates of the midpoint of a segment, you are
actually finding the average (mean) of the xcoordinates and the average
(mean) of the ycoordinates.
This concept of finding the average
of the coordinates can be written as a formula:
The Midpoint
Formula:
The midpoint
of a segment with endpoints (x_{1 }, y_{1})
and (x_{2 }, y_{2}) has
coordinates


NOTE:
The Midpoint Formula works for all
line segments: vertical, horizontal or diagonal. 

Find
the midpoint of line segment
.
A(3,4)
B(2,1)
The midpoint will have coordinates
Answer 
NOTE: Don't be surprised if
your answer contains a fraction. Answers may be left in
fractional form or written as decimals. 

Consider
this "tricky" midpoint problem: 

M is the midpoint
of
. The coordinates
M(1,1) and
C(1,3) are given.
Find the coordinates of point
D.
The
coordinates of
point
D are (3,5). 
First,
visualize the situation. This will give you an idea of
approximately where point
D will be located. When you find
your answer, be sure it matches with your visualization of where the
point should be located.
Solve algebraically:
M(1,1),
C(1,3) and
D(x,y)
Substitute into the Midpoint Formula:
Solve for each variable
separately: 


Other Methods of
Solution: 
Verbalizing the algebraic solution:
Some students like to do these
"tricky" problems by just examining the coordinates and asking
themselves the following questions:
"My midpoint's xcoordinate is 1. What is 1
half of? (Answer 2)
What do I add to my endpoint's xcoordinate of +1 to get 2? (Answer
3)
This answer must be the xcoordinate of the other endpoint."
These students are simply verbalizing the algebraic solution.
(They use the same process for the ycoordinate.)
Utilizing
the concept of slope and congruent triangles:
A line segment is part
of a straight line whose slope (rise/run) remains the same no matter where
it is measured. Some students like to look at the rise and run values
of the x and y coordinates and utilize these values to find the missing
endpoint.

Find
the slope between points
C and
M. This slope has a run of 2
units to the left and a rise of 4 units up. By repeating this
slope from point
M (move 2 units to the left and 4 units up), you
will arrive at the other endpoint.
By using this slope
approach, you are creating two congruent right triangles whose legs
are the same lengths. Consequently, their hypotenuses are also
the same lengths and
DM = MC making
M the midpoint of
.

