A midpoint splits a segment into two equal parts, but a segment can be divided in any ratio. This lesson partitions a directed line segment: given two endpoints and a ratio a : b, find the point that lies aa + b of the way from the first point to the second. The work is done with rise and run, one coordinate at a time, and the direction matters — the ratio is always measured from the starting point, the point named first.
To partition a segment is to divide it into parts with a given relationship. When the parts are equal, divide the total rise and the total run by the number of parts and add one step at a time, starting from the first point. When the parts stand in a ratio a : b, multiply the total rise and the total run by the fraction aa + b and add the pieces to the starting point's coordinates.
What you are looking at: Segment AB on a grid, with A(0, 0) and B(9, 6), cut into equal parts. The green points are the partition points, and the matching tick marks — one at the middle of each piece — show that the pieces are congruent (equal in measure).
What to do: Use the buttons under the diagram to choose how many equal parts to cut the segment into. A marker walks the steps from A to B, and the readout works the current number of parts in full.
What to notice: Every step adds the same amount — 1n of the total run and 1n of the total rise — and the steps always start at A, the first-named point.
Number of equal parts — choose how many equal pieces segment AB is cut into. The step points, the tick marks, and the readout all update.
What you are looking at: A directed segment on a grid, with the partition point P shown in green. The gold highlight covers the part from the starting point to P — the part the ratio measures — and the arrowhead at the far end shows the direction of the segment. The dashed lines show the total run and the total rise from the starting point.
What to do: Drag the orange handle on A and the purple handle on B to move the endpoints. Use the steppers to change the ratio, and use the direction buttons to choose which endpoint the segment is directed from. P slides to its new position whenever the ratio or the direction changes.
What to notice: The fraction of the way is always the first ratio number over the sum, and it is always measured from the starting point — so partitioning segment AB and partitioning segment BA give different points, except when the ratio is 1 : 1.
Ratio number a — the part of the ratio measured from the starting point.
3Ratio number b — the remaining part of the ratio, toward the ending point.
1Direction — which endpoint the directed segment starts from. The fraction is measured from that point.
What to do: Each problem gives you a directed segment, a ratio, and a point P to find. Work in three steps: (1) choose the equation to use to solve the problem — the fraction of the way, measured from the correct starting point, (2) find the run and the rise in the direction of the segment, and (3) report the partition point the question asks for. The gold highlight in the diagram marks the part the current step works on, and a value found in an earlier step is written at the top of the diagram. Each choice you pick moves a marker in the diagram to the point that choice describes, whether the choice is right or wrong, so you can see what the choice means; a missed choice is shown as a dashed ghost marker. A first miss gives a hint and another try, with the missed choice removed. A second miss on a step names the correct choice and explains it, and the activity then moves on. When all three steps are complete, the full worked solution appears. Press New Item for the next problem: the four problems take turns in order, and the problem number is shown with the step question.
Ready to practice? Work through these six questions, one at a time. If your first answer is wrong, you will get a hint and another try; the full explanation appears once you answer correctly, or after a second miss. Your score counts first-try answers.
Partitioning a Directed Line Segment — Mr. J Geometry. All activities on this page run entirely in your browser.