Reading the ratio back
A parked car is somewhere near a straight road.
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- P(9, 7) lies on segment AB.
- Gap from A to P: (8, 6).
- Gap from P to B: (4, 3).
- Both gaps give the ratio 2 : 1.
The rule
To find AP : PB, use the gaps. Find the gap from A to P. Find the gap from P to B. Compare the across gaps. Compare the up gaps. Both must give the same ratio. That is because P cuts AB in one ratio.
Take A(1, 1), P(9, 7), B(13, 10). The gap AP is (9 − 1, 7 − 1) = (8, 6). The gap PB is (13 − 9, 10 − 7) = (4, 3). Across, 8 : 4 is 2 : 1. Up, 6 : 3 is also 2 : 1. So AP : PB = 2 : 1.
Skip a pair of gaps that are both 0. A pair of zeros gives no ratio. Use the other pair. A vertical segment has both across gaps 0.
If the two ratios differ, P is not on the segment.
| Points | Gap A to P | Gap P to B | Ratio |
|---|---|---|---|
| A(1, 1), P(9, 7), B(13, 10) | (8, 6) | (4, 3) | 2 : 1 |
| A(2, 1), Q(8, 7), B(14, 9) | (6, 6) | (6, 2) | not on AB |
In row two, the across gaps are 6 and 6. That is 1 : 1. The up gaps are 6 and 2. That is 3 : 1. The two ratios disagree, so Q is not on AB. Q only looks as if it is between A and B.