Cutting a segment into thirds
A road trip is split into three equal legs. In the lab, drag B. Put the marked stop on the point asked.
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- A(2, 3) to B(11, 15).
- First stop: (5, 7).
- Second stop: (8, 11).
- Three equal hops reach B.
The rule
Cutting a segment into three equal parts needs two points. Those two points are the points of trisection. The jump is the change in x and y. From A, add one third of the jump, then two thirds.
For A(2, 3) and B(11, 15): the whole jump is (9, 12). Here 11 − 2 = 9 and 15 − 3 = 12. One third is (9 ÷ 3, 12 ÷ 3) = (3, 4). The first point is (2 + 3, 3 + 4) = (5, 7). The second is (2 + 6, 3 + 8) = (8, 11).
| From A | Point | Ratio AP : PB |
|---|---|---|
| One third | (5, 7) | 1 : 2 |
| Two thirds | (8, 11) | 2 : 1 |
| Three thirds | B(11, 15) | end |
Each hop is (3, 4), and its length is √(9 + 16) = 5. Three hops make 15, the whole length of AB.
Four equal parts need three points. Their ratios AP : PB are 1 : 3, 2 : 2 and 3 : 1. The middle one reduces to 1 : 1.
Start from B instead, and the same two points result. Their ratios swap: 1 : 2 becomes 2 : 1.