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Integer Function
Problem Statement
How do the coordinates of a midpoint of a segment relate to the coordinates of the endpoints of the segment?


Problem setup

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.

 

 

Plans to 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.

Investigation/Exploration 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 endpoint.

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!  COOL!

 

Extensions of the Problem

What about midpoints in 3 space?

Author & Contact
Michelle Houston
email me

 

 

 

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