Partitioning a Directed Line Segment
Overview
Partitioning divides a segment into pieces in a stated ratio — a midpoint is the simplest partition, splitting a segment into two equal parts. On the coordinate plane the work is done with rise and run: find the total rise and run from the starting point, take the fraction of each that the ratio calls for, and add those pieces to the starting point's coordinates. Because the segment is directed, the ratio is measured from the first-named point.
Vocabulary
- Partition
- To divide a segment into parts with a given relationship. A midpoint partitions a segment into 2 equal parts; other partitions divide it into more parts, equal or in a stated ratio.
- Directed Line Segment
- A segment with a direction, from its starting point to its ending point. Partition ratios are measured from the starting point, so reversing the direction changes the partition point.
- Ratio
- A comparison of two quantities. A partition ratio of a:b places the point a/(a + b) of the way from the starting point to the ending point.
- Rise and Run
- The vertical change (rise) and horizontal change (run) from one point to another. To partition, divide the total rise and run into the required fraction and add the pieces to the starting point.
Main Ideas
- For equal parts, divide both the rise and the run by the number of parts, then add the step repeatedly from the starting point.
- For a ratio a:b, do not simplify the rise/run fraction; multiply the total rise and run by a/(a + b).
- Always add the pieces to the STARTING point's coordinates — the point named first.
- Partition coordinates may be decimals; a ratio like 3:1 gives the point 3/4 of the way along the segment.
- Sanity check: the partition point always lies between the endpoints, closer to the endpoint with the larger ratio share.
Lesson Notes
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Interactive Activity
Partition a directed segment into equal parts or in a given ratio, from either direction.
Partitioning a Directed Line Segment — Interactive ActivityOpen ›
Practice Problems
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Download the practice worksheet for this lesson, work the problems, then check your work with the solutions.