point halfway between the endpoints of a line segment is called the midpoint.
A midpoint divides a line segment into two equal segments.
a midpoint of a line segment
is the point on that line segment that divides the segment two congruent
Geometry, there are several ways to determine the midpoint of a line
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
the midpoints and
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)
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 x-coordinates and the average
(mean) of the y-coordinates.
This concept of finding the average
of the coordinates can be written as a formula:
of a segment with endpoints (x1 , y1)
and (x2 , y2) has
The Midpoint Formula works for all
line segments: vertical, horizontal or diagonal.
the midpoint of line segment
The midpoint will have coordinates
NOTE: Don't be surprised if
your answer contains a fraction. Answers may be left in
fractional form or written as decimals.
this "tricky" midpoint problem:
M is the midpoint
. The coordinates
C(1,-3) are given.
Find the coordinates of point
D are (-3,5).
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.
Substitute into the Midpoint Formula:
Solve for each variable
Other Methods of
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 x-coordinate is -1. What is -1
half of? (Answer -2)
What do I add to my endpoint's x-coordinate of +1 to get -2? (Answer
This answer must be the x-coordinate of the other endpoint."
These students are simply verbalizing the algebraic solution.
(They use the same process for the y-coordinate.)
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
the slope between points
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