How do the coordinates of a midpoint
of a segment relate to the coordinates of the endpoints of the segment?
Here, I am being asked to give
an explanation for the relationship of the midpoint of a line segment
– the point in the exact center of a segment, a point of symmetry so
to speak – and the coordinates of this point.
First, I will explain what we
are talking about. Segments are
measurable portions of a line.
The midpoint, as I stated above, is the point that exactly divides
the segment in half (two equal parts).
When we are dealing with segments on the coordinate plane, every
point has a certain location, known as its coordinates. Below is an example of a coordinate
plane. Later you will see
another example done on the grid.
Solve/Investigate the Problem
In the beginning of my thoughts
on this problem, I thought, “use the formula, stupid!”
Then, I figured that this
is really not enough of an explanation.
Instead, I decided to use the
Geometer’s Sketch Pad to illustrate.
of the Problem
Below, is segment AB. Here, A and B are the endpoints of
my line segment.
Next, I will draw, well, take a
look… The blue dashed lines are “parallel” to each
Below, I will use GSP to show
the middle of segments AD and DB.
You can also see the coordinates of the
middle of each of the segments mentioned above.
Now if you look closely, at the
coordinates of each midpoint, you can see that the coordinates of the midpoint
of segment AB are the x coordinate of the horizontal line and the y
coordinate of the vertical line!
Extensions of the Problem
What about midpoints in 3 space?
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