CONCEPT 1 – Directed Line Segments

We can reference the same partition of a line segment by using the different endpoints of the directed segment.  The diagram below demonstrates how you can reference the same location using either endpoint of the line segment.  Note how the wording changes for these two descriptions.

CONCEPT 2 – Dilations

Partitioning a directed line segment can be done using dilation.  Let me do a quick review of some key concepts about dilation.  The first thing that I want to review and emphasize is that dilation is directly connected to slope.  When we use a scale factor of 2, we are actually performing 2 slopes starting from the center of dilation.  In other words, we do two runs and two rises to determine the new location.  When we use a scale factor of 3, we are actually performing 3 slopes – three runs and three rises to determine the image.

These diagrams demonstrate the relationship between the dilation scale factor and the number of slopes that we do to determine the image.  This relationship will be very helpful in partitioning a line segment.

VERY IMPORTANT -- In the first example you might notice that WE DID NOT REDUCE THE SLOPE!!!  The slope should not be reduced or altered in any way because those values not only represent the slope they also the distance and the direction to the point.  In the same context, in the second example we do not want to move the negative value around. Changing the negative would not affect the slope but it would definitely alter the direction.

CONCEPT 3 – Partitioning a Directed Line Segment

Partitioning a line segment means to divide it up into pieces.  To relate this to a dilation it means that we will be doing a reduction (0 < k < 1) so that the point will be on the segment.