Ratios of Directed Line Segments Calculator
Find a partition point from a ratio, or recover the ratio from a known collinear point.
Inputs
Live result
P lies 40% of the directed path from A to B.
Partition breakdown
Coordinate plot
The plot uses the same current coordinates as the result cards and workbook. The solid stroke is segment A→B; an external partition also shows the required extension.
Point detail table
| Point | x-coordinate | y-coordinate | Directed parameter t | Role |
|---|---|---|---|---|
| A | 1 | 2 | 0 | Start endpoint |
| P | 2.2 | 3.6 | 0.4 | Internal partition point |
| B | 4 | 6 | 1 | End endpoint |
The parameter form is P = A + t(B – A). Internal points have 0 < t < 1; external points have t < 0 or t > 1.
How to use the ratios of directed line segments calculator
What this calculator does
This calculator locates a point P that partitions the directed segment from A to B in a stated ratio m:n, or works backward from a known collinear point P to an equivalent AP:PB ratio. It supports both internal division, where P lies strictly between the endpoints, and external division, where P lies on the same line beyond A or B. The result is an exact coordinate-geometry identity subject only to displayed decimal rounding. It does not decide whether a diagram is drawn to scale, prove that unrelated points are collinear, or assign physical units; all coordinates and distances use the same unit system you supply.
When to use it
Use the calculator to check analytic-geometry homework, divide a map or design segment proportionally, locate a weighted point between two coordinates, or verify whether a supplied point lies at the intended internal or external ratio. The coordinate plot is especially useful for spotting reversed endpoint direction or an external point appearing on the wrong side.
How to calculate
- The calculator opens with the demonstration A(1, 2), B(4, 6), internal ratio 2:3. Its first result is P(2.2, 3.6), and the Excel workbook is immediately available.
- Set Solve for to Point P coordinates when the ratio is known, or to Ratio m : n when P is known. Choose Partition type as Internal or External.
- Replace the endpoint coordinates, then enter the editable pair: m (AP part) and n (PB part) in point mode, or P x-coordinate (pₓ) and P y-coordinate (pᵧ) in ratio mode. Results, plot, table, and workbook readiness update automatically.
- Read the partition point, distances, directed parameter, and AP/PB shares. Select Download Excel to export the current canonical values. Reset clears the demonstration and all numeric fields; Excel export remains unavailable until a complete valid set is entered again.
Input guide
Solve for is a required two-option selector. Point mode treats m and n as inputs and derives pₓ and pᵧ; ratio mode does the reverse and reports a distance-equivalent ratio. Partition type is also required. Internal is valid only when P is between A and B. External is valid only beyond an endpoint, and when you enter m and n directly they must be positive and unequal because m – n is the denominator.
A x-coordinate (x₁), A y-coordinate (y₁), B x-coordinate (x₂), and B y-coordinate (y₂) are required decimal numbers using a period as the decimal separator. Values such as -2, 0, and 4.75 are accepted; commas, scientific notation, and mixed-unit text are rejected rather than silently reinterpreted. A and B cannot be identical. Changing the endpoint order changes the directed parameter because the segment direction is always A→B.
m (AP part) and n (PB part) are positive, dimensionless ratio components in point mode. The sample 2 and 3 means AP:PB = 2:3. Multiplying both by the same positive factor does not change P. A common mistake is to swap m and n: m weights endpoint B in the section formula because P is m parts of the way from A, while n weights endpoint A. P x-coordinate (pₓ) and P y-coordinate (pᵧ) are required decimals in ratio mode. P must be collinear with A and B and must occupy the region selected by Partition type.
Output guide
Partition point gives P(pₓ, pᵧ). Segment length AB, Distance AP, and Distance PB are Euclidean lengths in the same coordinate unit. Directed parameter t comes from P = A + t(B – A): t between 0 and 1 is internal, while a negative value or a value above 1 is external. AP share and PB share express the two distance parts as percentages of AP + PB; they sum to 100%. Vector Δx and Vector Δy are x₂ – x₁ and y₂ – y₁. The summary pills repeat the active partition, solver target, equivalent ratio, and t value.
The Coordinate plot draws A, P, and B from the same model values. The solid line is segment A→B; a dashed continuation appears for external division. The Point detail table lists Point, x-coordinate, y-coordinate, Directed parameter t, and Role. A has t = 0 and B has t = 1 by definition. These are exact model relationships, not recommendations or statistical estimates.
Worked example
For A(1, 2), B(4, 6), and internal ratio 2:3, the x-coordinate is (2×4 + 3×1) ÷ (2+3) = 11÷5 = 2.2. The y-coordinate is (2×6 + 3×2) ÷ 5 = 18÷5 = 3.6. Therefore P = (2.2, 3.6). The endpoint difference is (3, 4), so AB = √(3²+4²) = 5. The parameter is t = 2÷5 = 0.4, giving AP = 0.4×5 = 2 and PB = 3. These values match the first-open result cards, plot, detail table, and exported workbook.
Learn more
The parameter representation used here is the same idea developed in OpenStax's discussion of parametric equations in two dimensions. For broader Cartesian plotting context, review graphs of linear functions. Paul's Online Notes also provides a concise reference on equations and properties of lines.
Internal and external section formulas
For internal division, P = ((m x₂ + n x₁)/(m+n), (m y₂ + n y₁)/(m+n)). This is a weighted average, so positive m and n keep P between A and B. For external division, P = ((m x₂ – n x₁)/(m – n), (m y₂ – n y₁)/(m – n)). The denominator explains why equal external ratio components are undefined: they would place the partition point at infinity rather than at a finite coordinate.
Interpreting direction correctly
A directed segment distinguishes A→B from B→A. Reversing the endpoints preserves the geometric line and total length, but changes the parameterization and which side counts as “beyond A” or “beyond B.” When a ratio result looks inverted, first confirm the endpoint order and whether m represents AP while n represents PB.