Verifying that a Midpoint Exists
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This activity will help to reinforce the idea that there is more than one way to solve a problem.

 

For the graph at the left, find at least two ways to verify that M is the midpoint of segment .  Show all work.  Be sure to "convince me" that your argument is valid.

A(-3,1)  M(0,2)  C(3,3)

 

Here are some possible answers:

Using the Distance Formula


Since the distance from A to M is the same as the distance from M to C, M is the midpoint of .

 Using the Midpoint Formula


The midpoint of is (0,2).  Since the coordinates of point M are (0,2), M is the midpoint of .


 

Using Slope and Congruent Triangles

The graph above shows two congruent right triangles whose corresponding legs are of equal length (obtained from counting).  Since the triangles are congruent, their hypotenuses are congruent,  and AM = MC.  Since M divides the segment into two equal parts, M is the midpoint of the segment.