The section formula locates a point a given fraction of the way along a segment – a centre of mass on a rod, a point on a map route, or the centroid of a triangle (which divides each median 2 : 1).
How to use the Ratios of Directed Line Segments Calculator
- Endpoint A – x
- Endpoint A – y
- Endpoint B – x
- Endpoint B – y
- The line is partitioned
- Ratio m
- Ratio n
Results update as you type, and everything is calculated in your browser.
How it works
For an internal division, P lies between A and B with AP : PB = m : n. For an external division, P lies on the extension of AB.
Inputs explained
| Field | What to enter |
|---|---|
| Endpoint A – x | A number |
| Endpoint A – y | A number |
| Endpoint B – x | A number |
| Endpoint B – y | A number |
| The line is partitioned | Choose one: Internally, Externally |
| Ratio m | A number |
| Ratio n | A number |
Example values
How the main result changes when Endpoint A – x changes and everything else stays at the example values above:
| Endpoint A – x | Point P – x |
|---|---|
| 0.5 | 6.875 |
| 0.75 | 6.9375 |
| 1 | 7 |
| 1.25 | 7.0625 |
| 1.5 | 7.125 |
| 2 | 7.25 |
Formula
Internal: P = ((m·x₂ + n·x₁)/(m + n), (m·y₂ + n·y₁)/(m + n))
External: P = ((m·x₂ − n·x₁)/(m − n), (m·y₂ − n·y₁)/(m − n))
Worked example
Inputs — Endpoint A – x: 1; Endpoint A – y: 2; Endpoint B – x: 9; Endpoint B – y: 6; The line is partitioned: Internally; Ratio m: 3; Ratio n: 1.
Results — Point P – x = 7; Point P – y = 5; Fraction of the way from A to B = 0.75.
Key terms
- Section formula
- The coordinates of a point dividing a segment in a ratio.
Frequently asked questions
What does a ratio of 3 : 1 mean?
P is three quarters of the way from A to B.
What is the midpoint ratio?
1 : 1.
What is an external division?
P lies outside AB with AP : PB = m : n.
Why can't m = n externally?
The formula divides by m − n = 0.