The section formula
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- A(1, 1) and B(13, 10).
- P(9, 7) divides AB.
- AP is 2 parts of the ratio.
- PB is 1 part of the ratio.
The rule
Counting hops works, but you must count again for every ratio. One formula does every ratio.
Section formula: the point P divides AB in the ratio m : n. Its coordinates are ((m·x₂ + n·x₁)/(m + n), (m·y₂ + n·y₁)/(m + n)). Here A is (x₁, y₁) and B is (x₂, y₂).
This means AP : PB = m : n, with P between A and B.
B's numbers carry m. A's numbers carry n. Both sit over m + n. When m is bigger than n, AP is longer. So P is nearer B, and B's numbers count more.
Take A(1, 1), B(13, 10) and the ratio 2 : 1. Here m = 2, n = 1 and m + n = 3. Across is (2×13 + 1×1)/3 = 27/3 = 9. Up is (2×10 + 1×1)/3 = 21/3 = 7. The point is (9, 7). Counting hops gave the same point.
| Ratio | Across | Up | Point |
|---|---|---|---|
| 2 : 1 | (2×13 + 1×1)/3 = 9 | (2×10 + 1×1)/3 = 7 | (9, 7) |
| 1 : 2 | (1×13 + 2×1)/3 = 5 | (1×10 + 2×1)/3 = 4 | (5, 4) |
| 1 : 1 | (1×13 + 1×1)/2 = 7 | (1×10 + 1×1)/2 = 11/2 | (7, 11/2) |
The ratio 1 : 1 gives the midpoint. It is the point halfway between A and B.
Watch out!
Pair m with A and you get the other cut. For 2 : 1 you get (5, 4). That is the cut 1 : 2.
Try the lab to finish this lesson.